Kinematics is defined as the study of motion without considering its causes. The word “kinematics” comes from a Greek term meaning motion and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion). In one-dimensional kinematics we will study only the motion of a ball, for example, without worrying about what forces cause or change its motion. Such considerations come in other chapters. In this chapter, we examine the simplest type of motion—namely, motion along a straight line, or one-dimensional motion.

Position

In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the positionof the rocket with respect to the Earth as a whole, while a professor’s positioncould be described in terms of where she is in relation to the nearby white board.  In other cases, we use reference frames that are not stationary but are in motionrelative to the Earth. To describe the position of a person in an airplane, for example, we use the airplane, not the Earth, as the reference frame.  

Displacement

If an object moves relative to a reference frame (for example, if a professor moves to the right relative to a white board or a passenger moves toward the rear of an airplane), then the object’s position changes. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced.

Definition: 

DISPLACEMENT

Displacement is the change in position of an object:

Δx=xf−x0,

where Δx is displacement , x_f is the final position and x ₀ is the initial position

In this text the upper case Greek letter Δ always means “change in” whatever quantity follows it; thus, Δx means change in position

We can find displacement by subtracting initial position X₀ from the final position X_f

Note that the SI unit for displacement is the meter (m), but sometimes kilometres, miles, feet, and other units of length are used. Keep in mind that when units other than the meter are used in a problem, you may need to convert them into meters to complete the calculation.

A professor paces left and right while lecturing. Her position relative to Earth is given by x. The +2.0 m displacement of the professor relative to Earth is represented by an arrow pointing to the right.

A passenger moves from his seat to the back of the plane. His location relative to the airplane is given by x. The −4.0 m displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. Notice that the arrow representing his displacement is twice as long as the arrow representing the displacement of the professor (he moves twice as far)  

Note that displacement has a direction as well as a magnitude. The professor’s displacement is 2.0 m to the right, and the airline passenger’s displacement is 4.0 m toward the rear. In one-dimensional motion, direction can be specified with a plus or minus sign. When you begin a problem, you should select which direction is positive (usually that will be to the right or up, but you are free to select positive as being any direction).

The professor’s initial position is X₀ =1.5 m and her final position is X_f =3.5 m . Thus her displacement is

Δx=Xf−X0=3.5 m−1.5 m=+2.0 m.

In this coordinate system, motion to the right is positive, whereas motion to the left is negative.

Similarly, the airplane passenger’s initial position is X0=6.0 m and his final position is Xf=2.0m and hence his displacement is

Δx=Xf−X0=2.0 m−6.0 m=−4.0 m.

His displacement is negative because his motion is toward the rear of the plane, or in the negative x direction in our coordinate system.

Distance

Although displacement is described in terms of direction, distance is not. 

Distance is defined to be the magnitude or size of displacement between two positions. Note that the distance between two positions is not the same as the 

distance travelled between them. 

Distance travelled is the total length of the path traveled between two positions. 

Distancehas no direction and, thus, no sign. For example, the distance

 the professor walks is 2.0 m. The distance the airplane passenger walks is 4.0 m.

MISCONCEPTION ALERT: 

DISTANCE TRAVELED VS. MAGNITUDE OF DISPLACEMENT

It is important to note that the distance traveled, however, can be greater than the magnitude of the displacement (by magnitude, we mean just the size of the displacement without regard to its direction; that is, just a number with a unit). For example, the professor could pace back and forth many times, perhaps walking a distance of 150 m during a lecture, yet still end up only 2.0 m to the right of her starting point. In this case her displacement would be +2.0 m, the magnitude of her displacement would be 2.0 m, but the distance she travelled would be 150 m.

In kinematics we nearly always deal with displacement and magnitude of displacement, and almost never with distance travelled. One way to think about this is to assume you marked the start of the motion and the end of the motion. The 

Displacement is simply the difference in the position of the two marks and is independent of the path taken in traveling between the two marks. The 

distance travelled, however, is the total length of the path taken between the two marks.

Exercise 

A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is her displacement? (b) What distance does she ride? (c) What is the magnitude of her displacement?

Answer

The rider’s displacement is .

a) Δx=Xf−X0 = 2 – 3 = -1  (The displacement is negative because we take east to be positive and west to be negative.)

 (b) The distance traveled is 3 km + 2 km = 5 km.

(c) The magnitude of the displacement is 1 km

Vectors, Scalars, and Coordinate Systems

What is the difference between distance and displacement?

Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity.

A vector is any quantity with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down. The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (−) sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector’s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector. Some physical quantities, like distance, either have no direction or none is specified.

A scalar is any quantity that has a magnitude, but no direction. For example, a 20ºC temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km/h speed limit, a person’s 1.8 m height, and a distance of 2.0 m are all scalars—quantities with no specified direction. Note, however, that a scalar can be negative, such as a −20ºC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.

Coordinate Systems for One-Dimensional Motion

In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however, as with the jet in Figure , it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it. Figure :

It is usually convenient to consider motion upward or to the right as positive (+) and motion downward or to the left as negative (−).

Time, Velocity, and Speed :

There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other concepts. In this section we add definitions of time, velocity, and speed to expand our description of motion.

Time

The most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple—time is change, or the interval over which change occurs. It is impossible to know that time has passed unless something changes. The amount of time or change is calibrated by comparison with a standard. The SI unit for time is the second, abbreviated s. We might, for example, observe that a certain pendulum makes one full swing every 0.75 s. We could then use the pendulum to measure time by counting its swings or, of course, by connecting the pendulum to a clock mechanism that registers time on a dial. This allows us to not only measure the amount of time, but also to determine a sequence of events.

How does time relate to motion? We are usually interested in elapsed time for a particular motion, such as how long it takes an airplane passenger to get from his seat to the back of the plane. To find elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, a lecture may start at 11:00 A.M. and end at 11:50 A.M., so that the elapsed time would be 50 min.

Elapsed time Δt is the difference between the ending time and beginning time,

Δt = t_f – t₀ , where

Where Δt is the change in time or elapsed time, is the time at the end of the motion, and t₀ is the time at the beginning of the motion.

Velocity

Your notion of velocity is probably the same as its scientific definition. You know that if you have a large displacement in a small amount of time you have a large velocity, and that velocity has units of distance divided by time, such as miles per hour or kilometres per hour.

Average velocity is displacement (change in position) divided by the time of travel,

ṽ =  Δx / Δt  = (X_f −X₀) /  (t_f −t₀)

where ṽ  is the average (indicated by the bar over the v) velocity, Δx is the change in position (or displacement), and X_f  and X₀ are the final and beginning positions at times t_f and t₀  respectively

Notice that this definition indicates that velocity is a vector because displacement is a vector. It has both magnitude and direction. The SI unit for velocity is meters per second or m/s, but many other units, such as km/h, mi/h (also written as mph), and cm/s, are in common use.

Suppose, for example, an airplane passenger took 5 seconds to move −4 m (the negative sign indicates that displacement is toward the back of the plane). His average velocity would be

ṽ =  Δx / Δt  = (-4m)/5s = -0.8 m/s

The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he goes to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals.

The smaller the time intervals considered in a motion, the more detailed the information. When we carry this process to its logical conclusion, we are left with an infinitesimally small interval. Over such an interval, the average velocity becomes the instantaneous velocity or the velocity at a specific instant. A car’s speedometer, for example, shows the magnitude (but not the direction) of the instantaneous velocityof the car. (Police give tickets based on instantaneous velocity, but when calculating how long it will take to get from one place to another on a road trip, you need to use average velocity.) 

Instantaneous velocity v is the average velocity at a specific instant in time (or over an infinitesimally small timeinterval).

Speed

In everyday language, most people use the terms “speed” and “velocity” interchangeably. In physics, however, they do not have the same meaning and they are distinct concepts. One major difference is that speed has no direction. Thus speed is a scalar . Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and average speed

Instantaneous speed is the magnitude of instantaneous velocity . For example, suppose the airplane passenger at one instant had an instantaneous velocity  of −3.0 m/s (the minus meaning toward the rear of the plane). At that same time  his instantaneous speed was 3.0 m/s. Or suppose that at one time during a shopping trip your instantaneous velocity is 40 km/h due north. Your instantaneous speed at that instant would be 40 km/h—the same magnitude but without a direction. 

Average speed , however, is very different from average velocity.  Average speed  is the distance travelled  divided by elapsed time .

We have noted that distance travelled  can be greater than displacement. So average speedcan be greater than average velocity , which is displacement  divided by time . For example, if you drive to a store and return home in half an hour, and your car’s odometer shows the total distance travelled  was 6 km, then your average speed  was 12 km/h. Your average velocity , however, was zero, because your displacement  for the round trip is zero. (Displacement is change in position  and, thus, is zero for a round trip.) Thus average speed is not simply the magnitude of average velocity

Another way of visualizing the motion  of an object is to use a graph. A plot of position or of velocity as a function of time  can be very useful. For example, for this trip to the store, the position , velocity, and speed-vs.-time graphs are displayed (Note that these graphs depict a very simplified model of the trip. We are assuming that speed is constant during the trip, which is unrealistic given that we’ll probably stop at the store. But for simplicity’s sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store and the house is a perfectly straight line.)

.

Position vs. time , velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative.

Acceleration :

In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration, the greater the change in velocity over a given time . The formal definition of 

Acceleration is consistent with these notions, but more inclusive.

AVERAGE ACCELERATION

Average Acceleration is the rate at which velocity changes,

ặ=Δv / Δt=(V_f−V₀) / (T_f−T₀)

where ặ  is average acceleration ,V is velocity, and T is time

Because acceleration is velocity in m/s divided by time  in s, the SI units for acceleration are m/s ²   (meters per second  squared)

Recall that velocity is a vector —it has both magnitude and direction. This means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a car turns a corner at constant speed, it is accelerating because its direction is changing. The quicker you turn, the greater the acceleration. So there is an Acceleration when velocity changes either in magnitude (an increase or decrease in speed) or in direction, or both.

ACCELERATION AS A VECTOR

Acceleration is a  vector in the same direction as the change in velocity, Δv. Since velocity is a vector , it can change either in magnitude or in direction. Acceleration  is therefore a change in either speed or direction, or both. Keep in mind that although acceleration  is in the direction of the change in velocity, it is not always in the direction of motion

. If acceleration  is in a direction opposite to the direction of motion , the object slows down.

Problem :

A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration ?

First we draw a sketch and assign a coordinate system to the problem. This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have negative velocity.

We can solve this problem by identifying Δv and Δt from the given information and then calculating the average acceleration directly from the equation

ặ=Δv / Δt=(V_f−V₀) / (T_f−T₀)

We know that

V₀=0, V_f =−15.0 m/s (the negative sign indicates direction toward the west), Δt=1.80 s

Since the horse is going from zero to −15.0 m/s, its change in velocity equals its final velocity: Δv=v_f=−15.0 m/s

Substituting the values , we  get

ặ  = (−15.0 m/s) / 1.80 s= −8.33 m/s²

Falling Objects

Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process.

Gravity

The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the same constant acceleration, independent of their mass. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones.

In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time . (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction  between objects—such as between clothes and a laundry chute or between a stone and a pool into which it is dropped—also opposes motion between them. For the ideal situations of these first few chapters, an object falling without air resistance or friction  is defined to be in free-fall . The forceof gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called the acceleration due to gravity . The acceleration due to gravity is constant, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so important that its magnitude is given its own symbol, g. It is constant at any given location on Earth and has the average value

g=9.80 m/s²

The direction of the acceleration due to gravity  is downward (towards the center of Earth). In fact, its direction defines what we call vertical. Note that whether the acceleration a in the kinematic equations has the Value +g or −g depends on how we define our coordinate system . If we define the upward direction as positive, then a=−g=−9.80 m/s² , and if we define the downward direction as positive, then a=g=9.80 m/s²

KINEMATIC EQUATIONS FOR OBJECTS IN FREE FALL :

V=V₀−gt

y=y₀+v₀ t−1/2(gt²)

v²=(v₀)²−2g(y−y₀)

Example problem :

A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m/s. The rock misses the edge of the cliff as it falls back to earth. Calculate the position  and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance .

Solution :

Draw a sketch

We are asked to determine the position y at various times. It is reasonable to take the initial position y₀ to be zero. This problem involves one-dimensional Motion  in the vertical direction. We use plus and minus signs to indicate direction, with up being positive and down negative. Since up is positive, and the rock is thrown upward, the initial velocity must be positive too. The acceleration due to gravity  is downward, so a is negative. It is crucial that the initial velocity and the acceleration due to gravity  have opposite signs. Opposite signs indicate that the acceleration due to gravity  opposes the nitial motion and will slow and eventually reverse it.

Since we are asked for values of position  and velocity at three times, we will refer to these as y₁ and v₁ ; y₂ and v₂; and y ₃ and v₃

Solution for Position y₁ :

We know that y₀=0; v₀=13.0 m/s; a=−g=−9.80 m/s² and t=1.00 s.

We need to Identify the best equation to use. We will use 

y=y₀+v₀ t−1/2(gt²)

which is the value we want to find.

Substituting the known values and solve for y1

y1=0+(13.0 m/s)(1.00 s) – 1/ 2(−9.80 m/s2)(1.00 s)² =8.10 m

Solution for Velocity V₁ :

Given that 

y₀=0;  v₀=13.0 m/s;  a=−g=−9.80 m/s²

We also know from the solution above that y1=8.10 m

The best equation to use is v=v₀−gt

 Where (−g )= gravitational acceleration

Substituting the values , we get  

V₁=v₀ −gt=13.0 m/s−(9.80 m/s²)(1.00 s)=3.20 m/s

Solution for Remaining Times

The procedures for calculating the position and velocity att = 2.00 s and 3.00 s are the same as those above. The results are summarized in the Table below :

Time , t	Position , y	Velocity, v
1.00 s	8.10 m	3.20 m
2.00 s	6.40 m	−6.60 m
3.00 s	−5.10m	−16.4 m

Centripetal Acceleration :

We defined acceleration  as a change in velocity, either in its magnitude or in its direction, or both. When an object moves along a circular path, the direction of its velocity changes constantly, so there is always an associated acceleration, even if the speed of the object is constant. You experience this acceleration  yourself when you turn a corner in your car. What you notice is a sideways acceleration  because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration  will become. In this section we briefly examine the direction and magnitude of that acceleration.

The below image shows an object moving in a circular path at constant speed, called uniform circular motion . The direction of the instantaneous velocity  is shown at two points along the path. Acceleration  is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. We call the acceleration of an object moving in uniform circular motion  (resulting from a net external force) the centripetal acceleration (a_c ) – centripetal means “toward the center” or “center seeking.”

The directions of the velocity of an object at two different points are shown, and the change in velocity Δv is seen to point directly toward the center of curvature. (See small inset.) Because ac=Δv/Δt , the acceleration is also toward the center; ac is called centripetal acceleration. (For small time differences, Δθ is very small, and the arc length Δs is approximately equal to the chord length Δr)

The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? If we use the geometry shown in the above Figure along with some kinematics equations, we can obtain

ac=v² / r

which is the acceleration of an object in a circle of radius r at a speed v.

We see in the above Equation that centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that ac is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km/h than at 50 km/h. A sharp corner has a small radius, so that a_c is greater for tighter turns .

The unit of centripetal acceleration is m/s²

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Introduction

Physics is a fascinating branch of science that seeks to understand the fundamental principles governing the behaviour of the universe. One important aspect of physics is the study of physical quantities and their dimensions. In this lesson, we will explore the concept of dimensions, delve into dimensional analysis, and discover its practical application

Dimensions of Physical Quantities

1.1 Definition:
Physical quantities are measurable properties or characteristics of objects or phenomena, such as length, time, mass, and Velocity

Dimensions represent the nature of these quantities and provide a framework for their measurement
1.2 Fundamental Dimensions :

There are seven fundamental dimensions in the International System of Units (SI): length (L), mass (M), time (T), electric current (I), temperature (Θ), amount of substance (N), and luminous intensity (J)

All other physical quantities can be derived from these fundamental dimensions

Quantity	Unit	Symbol	Definition
Length	Meter	m	The meter is the distance traveled by light in a vacuum in 1/299,792,458th of a second.
Mass	Kilogram	kg	The kilogram is defined as the mass of the International Prototype of the Kilogram, a platinum-iridium cylinder kept at the International Bureau of Weights and Measures.
Time	Second	s	The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.
Electric Current	Ampere	A	The ampere is the constant current that, if maintained in two parallel conductors of infinite length and negligible cross-sectional area, placed 1 meter apart in a vacuum, would produce a force between the conductors of exactly    2 x 10-7 newton per meter of length.
Temperature	Kelvin	K	The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
Amount of Substance	Mole	mol	The mole is the amount of a substance that contains as many elementary entities (such as atoms, molecules, ions, or particles) as there are atoms in exactly 0.012 kilograms of carbon-12.
Luminous Intensity	Candela	cd	The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

These fundamental units provide a standardized and universally accepted system of measurement for various physical quantities, allowing for consistency and accuracy in scientific calculations and communications.

Advantages of SI Units

The SI (International System of Units) unit system offers several advantages:

1. Universality: The SI unit system is widely recognized and accepted globally. It provides a consistent and unified framework for measurements across different countries and scientific disciplines. This universality promotes international collaboration, communication, and understanding in various fields of study.
2. Standardization: SI units have well-defined definitions and are based on fundamental constants of nature. This standardization ensures accuracy, reliability, and reproducibility in measurements. It allows for precise comparisons and consistent results in scientific experiments and observations.
3. Coherence: SI units are designed to be coherent, meaning they are interrelated in a logical and consistent manner. The coherent system of units simplifies calculations and conversions between different quantities, as they are based on consistent mathematical relationships.
4. Scalability: The SI unit system is scalable, meaning it provides prefixes that can be used to express measurements across a wide range of magnitudes. This scalability makes it convenient to work with both extremely large and small values without the need for excessive zeros or complex conversions.
5. Compatibility with Scientific Laws: SI units are compatible with the laws of physics and other scientific principles. They align with the fundamental theories and principles that govern the behavior of the physical world, allowing for the formulation and testing of scientific theories, laws, and equations.
6. Practicality: The SI unit system is designed to be practical and user-friendly. The units are intuitive and easily understandable, facilitating effective communication and comprehension of measurements and quantities among scientists, engineers, and the general public.
7. International Recognition: The SI unit system is officially recognized by international organizations such as the International Bureau of Weights and Measures (BIPM), ensuring consistent usage and adoption in scientific research, education, industry, and trade worldwide.
   These qualities make it an indispensable tool for precise and accurate measurements in scientific and technological advancements.

1.3 Derived Dimensions

Derived dimensions are obtained by combining the fundamental dimensions using multiplication, division, or exponentiation
For example, velocity has dimensions of [L][T]⁻¹, where [L] represents length and [T]⁻¹ represents the inverse of time
Here is a table of 22 derived SI units along with the basic units involved in their derivation:

Derived SI Unit	Quantity	Definition	Basic Units Involved
Newton (N)	Force	1 N = 1 kg * m/s2	kilogram (kg), meter (m), second (s)
Pascal (Pa)	Pressure	1 Pa = 1 N/m2	Newton (N), meter (m)
Joule (J)	Energy	1 J = 1 N * m = 1 kg * m2/s2	Newton (N), meter (m), second (s)
Watt (W)	Power	1 W = 1 J/s = 1 kg * m2/s3	Joule (J), second (s)
Coulomb (C)	Electric charge	1 C = 1 A * s	Ampere (A), second (s)
Volt (V)	Electric potential	1 V = 1 W/A = 1 kg * m2/(A * s3)	Watt (W), Ampere (A), second (s)
Ohm (Ω)	Electrical resistance	1 Ω = 1 V/A = 1 kg * m2/(A2 * s3)	Volt (V), Ampere (A), second (s)
Farad (F)	Capacitance	1 F = 1 C/V = 1 A*s/V =            1 kg-1 * m-2 * s4 * A2	Coulomb (C), Volt (V), Ampere (A), second (s), kilogram (kg), meter (m)
Tesla (T)	Magnetic flux density	1 T = 1 Wb/m^2 = 1 kg/(A * s2)	Weber (Wb), meter (m), Ampere (A)
Henry (H)	Inductance	1 H = 1 V * s/A = 1 kg * m2/(A2 * s2)	Volt (V), second (s), Ampere (A)
Hertz (Hz)	Frequency	1 Hz = 1/s	1/second (1/s)
Siemens (S)	Electrical conductance	1 S = 1 A/V = 1 s3 * A2/(kg * m2)	Ampere (A), Volt (V), second (s), kilogram (kg), meter (m)
Weber (Wb)	Magnetic flux	1 Wb = 1 V * s = 1 kg * m2/(A * s2)	Volt (V), second (s), kilogram (kg), meter (m), Ampere (A)
Lux (lx)	Illuminance	1 lx = 1 lm/m2 = 1 cd * sr/m2	Lumen (lm), meter (m), candela (cd), steradian (sr)
Becquerel (Bq)	Radioactivity	1 Bq = 1/s	1/second (1/s)
Gray (Gy)	Absorbed dose	1 Gy = 1 J/kg	Joule (J), kilogram (kg)
Sievert (Sv)	Equivalent dose	1 Sv = 1 J/kg	Joule (J), kilogram (kg)
Weber per square meter	Magnetic field strength	1 Wb/m2 = 1 T	Weber (Wb), meter (m)
Radian (rad)	Plane angle	1 rad = 1 m/m	Meter (m)
Steradian (sr)	Solid angle	1 sr = 1 m2/m2	Meter (m)
Celsius (°C)	Temperature	°C = (K – 273.15)	Kelvin (K)

Prefixes and Multiples of SI Units

Since the magnitude of the SI units vary over a wide range , multiples and sub multiples are used to explain the units more precisely

Here is a table of the standard prefixes for SI units, including both multiples and submultiples

Prefix	Symbol	Multiplication Factor
Yotta	Y	1024
Zetta	Z	1021
Exa	E	1018
Peta	P	1015
Tera	T	1012
Giga	G	109
Mega	M	106
Kilo	K	103
Hecto	H	102
Deca	da	101
–	–	100 (Base Unit)
Deci	D	10-1
Centi	C	10-2
Milli	M	10-3
Micro	µ	10-6
Nano	N	10-9
Pico	P	10-12
Femto	F	10-15
Atto	A	10-18
Zepto	z	10-21
Yocto	y	10-24

Additional Practical Units of the SI system

Here are some practical units of the SI system

Parsec: The parsec (pc) is a unit of length used in astronomy to measure vast distances to stars and galaxies. It is approximately equal to 3.09 x 10¹⁶ meters.

Light Year: The light-year (ly) is another unit of astronomical distance. It represents the distance light travels in one year, and it is about 9.46 x 10¹⁵ meters.

Astronomical Unit: The astronomical unit (AU) is a unit of length used in astronomy to describe distances within our solar system. It is approximately the mean distance from the Earth to the Sun and is about 1.5 x 10¹¹ meters.

Micron: The micron (μm) is a unit of length equal to one millionth of a meter. It is often used to measure small distances, such as the width of a human hair or microscopic objects.

Angstrom: The angstrom (Å) is a unit of length used to measure atomic and molecular scales. It is equal to 0.1 nanometers or 10^-10 meters.

Fermi: The fermi (fm) is a unit of length used in nuclear and particle physics to measure atomic and subatomic scales. It is equal to 10^-15 meters.

X-ray Unit: The X-ray unit (XU) is a unit used to measure the intensity of X-rays. It is a non-SI unit and is defined as the X-ray exposure that produces 2.58 x 10^-4 coulombs per kilogram of air.

Atomic Mass Unit (amu): The atomic mass unit (amu) is a unit used to express the mass of atoms and molecules on a scale relative to the mass of a carbon-12 atom. 1 amu is approximately 1.66 x 10^-27 kilograms.

These units are used in various scientific and practical contexts, especially in astronomy, physics, and other fields dealing with very large or very small distances and masses.

Methods of Measurement of Physical Quantities

The direct method and indirect method are two approaches used to measure physical quantities. Here’s a brief explanation of each method:

Direct Method: In the direct method of measurement, the physical quantity of interest is measured directly using appropriate instruments or techniques. The measurement is made by comparing the quantity being measured to a known standard or using a calibrated instrument. This method provides a straightforward and accurate measurement of the desired quantity. For example, measuring the length of an object using a ruler or measuring the temperature using a thermometer are examples of direct measurements.

Indirect Method: In the indirect method of measurement, the physical quantity of interest is derived or calculated by measuring other related quantities and using mathematical relationships or equations. This method involves measuring multiple quantities and utilizing mathematical models or formulas to determine the desired quantity indirectly. The indirect method is used when direct measurement of the quantity is not feasible or when it is more practical to measure other related quantities. For example, determining the velocity of an object by measuring its displacement and time, or calculating the area of an irregular shape by measuring its dimensions and applying appropriate formulas are examples of indirect measurements.

Both direct and indirect methods have their advantages and limitations. The choice of method depends on the nature of the physical quantity being measured, the available instruments or techniques, and the accuracy required for the measurement. In many cases, a combination of direct and indirect methods may be employed to obtain accurate and reliable measurements of complex physical quantities

Let us consider the measurement of the Length of an object . If the object is measureable by placing a scale , then its called the Direct Method of measurement . However this method is not feasible if we measure the height of a hill , for which we need to adapt indirect methods

Various methods of measurement of Length

Echo Method of Length Measurement :

The echo method of length measurement is a technique used to determine the distance to an object or surface by measuring the time it takes for a sound wave or pulse to travel to the object and back. This method is commonly employed in various applications, such as sonar systems, ultrasound imaging, and distance measurement devices.

Here’s how the echo method works:

A sound wave or pulse is generated and emitted towards the target object or surface.

The sound wave travels through the medium (such as air or water) until it reaches the object.

Upon reaching the object, the sound wave reflects or echoes back towards the source.

A receiver detects the returning sound wave or echo.

The time elapsed between the emission of the sound wave and the reception of the echo is measured.

The distance to the object is then calculated using the known speed of sound in the medium and the measured time.

Since the speed of sound in a given medium is relatively constant, knowing the time it takes for the sound wave to travel allows for the calculation of the distance based on the equation:

Distance = (Speed of Sound × Time) / 2

The division by 2 is necessary because the measured time accounts for the round trip of the sound wave.

The echo method can be used to measure distances accurately over a wide range, from short distances in medical ultrasound imaging to large distances in sonar systems for navigation or oceanographic research. It is a non-contact measurement method that is often preferred when direct physical contact with the object being measured is not possible or practical.

RADAR method of Length measurement :

The radar method of length measurement is a technique that utilizes radar (radio detection and ranging) to determine the distance to an object or surface. It is widely used in applications such as navigation, weather monitoring, and remote sensing.

Here’s how the radar method works:

A radar system emits short pulses of electromagnetic waves, typically radio waves or microwaves.

These waves travel at the speed of light and are directed towards the target object or surface.

Upon encountering the object, a portion of the electromagnetic waves is reflected back towards the radar system.

The radar system’s receiver detects the reflected waves, commonly referred to as radar echoes.

The time delay between the emission of the pulse and the reception of the echo is measured.

The distance to the object is then calculated using the known speed of light and the measured time delay.

The speed of light is approximately 299,792,458 meters per second (in a vacuum). By multiplying the speed of light by the measured time delay and dividing it by 2 (since the time accounts for the round trip), the radar system can accurately determine the distance to the target.

Radar systems can provide precise and reliable distance measurements over various ranges, from short distances to long distances. Additionally, radar can also provide additional information about the target, such as its velocity or size, by analyzing the Doppler shift or the strength of the radar echoes.

The radar method is widely used in applications such as air traffic control, maritime navigation, meteorology, and even in automotive systems like radar-based distance sensors for collision avoidance. It allows for non-contact measurement and provides valuable information in a wide range of scenarios.

SONAR method of Length measurement :

The sonar method of length measurement is a technique that uses sound waves to measure the distance between an object or surface and the source of the sound. Sonar (Sound Navigation And Ranging) is commonly used underwater to determine distances, map the ocean floor, and detect underwater objects. Here’s an overview of how the sonar method works:

Sound Wave Generation: A sonar system emits sound waves, usually in the form of short pulses, into the water or another medium. The sound waves can be produced by specialized sonar transducers

Sound Wave Propagation: The emitted sound waves travel through the medium, such as water, until they encounter an object or surface. The sound waves propagate through the medium at a known speed, which is typically slower than the speed of light.

Reflection or Echo: When the sound waves encounter an object or surface, a portion of the sound energy is reflected back towards the sonar system. This reflection is referred to as an echo.

Detection: The sonar system’s receiver detects the returning echoes. The receiver is designed to capture and analyze the reflected sound waves.

Time Measurement: The time delay between the emission of the sound wave and the reception of the echo is measured precisely. This time delay is often referred to as the “echo time” or “round-trip time.”

Distance Calculation: The distance between the sonar system and the object or surface is then calculated using the known speed of sound in the medium and the measured echo time. The distance is determined by multiplying half of the echo time by the speed of sound:

Distance = (Speed of Sound × Echo Time) / 2

Sonar is widely used in various applications, including underwater navigation, bathymetry (measurement of water depth), fish finding, underwater imaging, and locating underwater structures or objects. It allows for non-contact measurement over long distances underwater, providing valuable information about the environment beneath the surface

LASER method of Length measurement :

The laser method of length measurement utilizes laser technology to accurately measure distances with a high degree of precision. This method is widely employed in various fields, including engineering, surveying, manufacturing, and scientific research. Here’s an overview of how the laser method works:

Laser Emission: A laser emits a highly focused beam of light, which travels in a straight line.

Target Reflection: The laser beam is directed towards the target object or surface. When the laser beam encounters the target, a portion of the light is reflected back towards the laser source.

Detection: The reflected laser light is detected using a sensor or detector. The detector captures the returning light and converts it into an electrical signal.

Time Measurement: The time taken for the laser light to travel to the target and back is precisely measured using electronic timing equipment. This time measurement is usually accomplished using high-frequency electronic circuits or time-of-flight measurement techniques.

Speed of Light: The speed of light in air or a vacuum is a well-known constant (approximately 299,792,458 meters per second). By multiplying the speed of light by the measured time and dividing it by 2 (since the time accounts for the round trip), the distance to the target can be calculated.

Distance = (Speed of Light × Time) / 2

Data Processing: Multiple measurements are often taken to improve accuracy and account for any variations or errors. Advanced data processing techniques, including statistical analysis and error correction algorithms, may be employed to enhance the precision of the measurements.

The laser method of length measurement offers numerous advantages, including non-contact measurement, high accuracy, and rapid data acquisition. It is commonly used for tasks such as distance ranging, 3D scanning, alignment, surface profiling, and dimensional inspection. Additionally, the laser method can be applied in both short-range and long-range measurement scenarios, depending on the specific laser technology and setup used.

So we see that multiple indirect measurement methods are available for all practical purposes.

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