Reference Point

To understand the concept of rest and motion, we need to understand the concept of reference point. The reference point is a fixed point on the basis of which we identify (refer) whether or not the objects are in rest or motion. If a body changes its position with respect to the reference point, the body is said to be in motion. Similarly, if a body does not change its position with respect to the reference point the body is said to be at rest.

From the examples above it is clear that rest and motion are relative terms. There is no such thing as universal rest or universal motion. Rest or motion is always with respect to some reference point.

Uniform & Non-uniform Motion

Suppose a car travels 40 km per hour. It means in every hour, the distance traveled by car is 40 km. It does no travel more or less than 40 km in one hour (per hour). This kind of motion is called a uniform motion. Therefore, the motion of a body that travels an equal distance in an equal interval of time is called uniform motion.

Sometimes, an object moves faster while sometimes the same object moves slower. It is not necessary that objects always move uniformly. It means that objects may not always travel equal distances in equal intervals of time. This kind of motion is called a non-uniform motion. Therefore, the motion of the body that cannot travel an equal distance in an equal interval of time is called non-uniform motion. This kind of non-uniform can be experienced generally when we are walking, running, driving, etc.

Speed and Velocity

Objects may travel fast or slow. For example, we may observe that an airplane travels very fast while a bicycle is slow. What, exactly do we mean when we use the term 'fast' and 'slow'? We are talking about the speed of an object when we mean slow or fast. Speed is the total distance traveled by a body per unit of time. In other words, it is the rate of change of distance. For example. if a car travels 160 km in 4 hours time then using a unitary method from math, the car moved 40 km every 1 hour. This means that the speed of the car was 40km per hour or 40km/hr. The SI unit of speed is m/s and the CGS unit is cm/s. It is a scalar quantity.

\(Speed(V)=\frac{Distance\;Traveled\;(S)}{Time\;Taken\;(t)}\)

\(\therefore V=\frac st\)

The shortest distance between any two points is called displacement. It is the distance in the straight line with a fixed direction. The rate of change of displacement of a body per unit time is called its velocity. i.e,

$$velocity(V)=\frac{Displacement(s)}{time(t)}$$

$$\therefore V=\frac st$$

The SI unit of velocity is m/s and CGS unit is cm/s. Veclocity is a vector quantity.

The speed and velocity means the same if body is moving in a straight line in a fixed direction. But if the body is moving in a curved path, it is better to say speed rather than velocity.

The ration of the total displacement covered by a body and the total time taken by the body is called the average velocity. (like the average/mean from math)

$$Average\;Velocity\;(V_a)=\frac{Displacement(s)}{time(t)}$$

$$\therefore v_a=\frac st$$ ------- (1)

Mathematically, the average velocity is the sum of initial velocity (u) and the final velocity (v), divided by 2.

\(\therefore v_a=\frac{u+v}2\) --------- (2)

Where, u = initial velocity
and, v = final velocity.

From equation (1) and (2) we get,
\(\frac st=\frac{u+v}2\)

Hence, \(\Rightarrow s=\frac{u+v}2\times t\) --------- (3)

This shows that a distance covered is equal to the product of average velocity and time taken.

Speed is measured by using an instrument called a speedometer.

Relative Velocity

The velocity of a body with respect to another body or a reference point is called relative velocity. For example, is hour house moving? The answer is generally no, our house is at rest. That means the speed of our house is zero. But This is true only if we compare our house with respect to the earth (ground). If you take the moon as a reference point, then the house will be in motion!! And the house now has a certain velocity. This very concept of velocity is known as relative velocity. Another common example would be a passenger is the bus are at rest relative to the bus but are in motion relative to the surrounding outside.

To realize the concept of relative velocity mathematically, we consider the following cases of two moving cars.

1. Same Direction-Same Speed

The relative of two bodies along the same direction withthe same velocity.

2. Same Direction-Different Speed

Relative velocity of two bodies along the same direction but with different velocities.

Let car A is moving with the velocity 30 m/s and car B is moving with the velocity 50 m/s in the same direction.

The figure shows the situation after 1 second. Car A has covered 30 m and car B has covered a distance of 50 m. Therefore, looking from the car A, the car B is appeared to be moving with the velocity.

\(V_{BA}=V_B-V_A=50-30=20\)

And from the car B, car A is appeared to be moving with the velocity,

\(V_{AB}=V_A-V_B=30-50=-20\)

3. Different Directions & Speed

The relative velocity of two bodies along the different directions with the same or different speed.

If a car A and B are moving in opposite directions with the same of different velocities then its situation after 1 second is shown in the figure below.

Therefore looking out from either of the cars, another car appears to be moving with the velocity

\(V_{AB}=V_A+V_B=25+40=65\) m/s

So, the cars appear to be moving with the velocity greater than their own velocities.

Acceleration

Suppose a body is increasing its velocity from 100 m/s to 200 m/s in 2 seconds. This is said to be the acceleration of the body. In this case, the body increases its velocity by 200 / 100 = 2 m/s in every second. This is called the acceleration of the body 2 m/s²

Therefore, acceleration is defined as the rate of change of velocity in per unit time. If 'v' is the final velocity changed from the initial velocity 'u' in the time 't'. Then acceleration 'a' is given by

\(a=\frac{v-u}t\)

Its SI unit is m/s2 and CGS unit is cm/s2. Acceleration is a vector quantity as it has both magnitudes as well as direction. If the final velocity of a body is less than its initial velocity, it is called negative acceleration or retardation. It is represented by '-a'.

NOTE:  A body in uniform motion (same velocity over time) doesn't have acceleration. As v = u such that, \(a=\frac{v-u}t=\frac{v-v}t=\frac0t=0\) m/s

 

Equations of Motion

In this section we deal with the equation of motion in a straight line. They are mathematical equations showing the relationship between initial velocity, finalvelocity, time taken, distance covered and acceleration. They are used to describe then motion of a body in a straight line. They are based on the definition of acceleration and definition of average velocity.

1. First Equation of Motion.

Suppose a body 'A' is moving with the velocity 'u'. After certain time taken 't', its velocity becomes 'v' and covers the distance 's'. During this, the body produces an acceleration 'a'.

Since acceleration is the rate of change of velocity,

We have, acceleration, \(a=\frac{v-u}t\),

\(\Rightarrow v=u+at\)

This is the first equation of motion in a straight line.

2. The Second Equation of Motion

We have, average velocity = \(\frac{v+u}t\)

Distance covered (s) = Average velocity x time

\(\Rightarrow s=\left(\frac{u+v}2\right)\times t\)
\(\Rightarrow s=\left(\frac{u+u+at}2\right)\times t\) \(\lbrack\because v=u+at\rbrack\) using first equation
\(\Rightarrow s=\frac{2ut+at^2}2\)
\(\Rightarrow s=\frac{2ut}2+\frac12at^2\)

\(\therefore s=ut+\frac12at^2\)

This is the second equation of motion of a body in a straight line.

3. The Third Equation of Motion

Starting from the first equation of motion in straight line. we have,
\(v=u+at\) -------- (i)

Suqaring both we get,

\(v^2=\left(u+at\right)^2\)
Or, \(v^2=u^2+2uat+a^2t^2\)
Or, \(v^2=u^2+2u\left(ut+\frac12at^2\right)\)

\(\therefore v^2=u^2+2as\) [ using the second equation of motion, replacing for s]

This is the third equation of motion of a body in a straight line.

4. Some Special Cases

• If a body changes its position with respect to the surrounding objects, then the body is said to be in motion.
• If a body does not change its position with respect to the surrounding objects, then the body is said to be at rest.
• The fixed point, about which we describe the motion of a body is called the reference point.
• The physical quantity having only magnitude but not a fixed direction is called a scalar quantity.
• The physical quantity having both magnitude and direction is called a vector quantity.
• Distance is the total length between two points. Its SI unit is m.
• Displacement is the shortest between the initial and final points. Its SI unit is also a meter.
• The rate of the change of distance is called speed and the rate of change of the displacement is called the velocity.
• Speed is a scalar quantity but velocity and acceleration is a vector quantity.
• If a body covers an equal distance in an equal interval of time, then the motion of the body is called uniform motion.
• If a body doesn't cover an equal distance in an equal interval of time, then the motion of the body is called a variable or non-uniform motion.
• The motion of a body relative to another object or the reference frame is called relative motion.