Velocity and Acceleration

In this chapter velocity and acceleration, it contains the concept of rest and motion, speed and velocity, relative velocity and numerical, etc. 

Rest and Motion

If a body is doesnot change its position with respect to other objects in its surrounding, then the body is said to be at rest. For example; a chair placed near a tree is at rest as here is no change in the position of chair with respect to the tree.

If a body changes its position with respect to others in its surrounding then the body is said to be in motion. For example; a car running on the road is in motion as it changes its position with respect to surrounding objects (tress, buildings on the road side).

 Rest and motion are relative terms: 

For determining the rest and motion of a body, we have to compare the position of the body with a certain stationary object. This stationary object is called reference point or reference frame. Thus, a stationary object with resect to which rest and motion state of a body is determined is called reference point or refernece frame. When we are in a moving bus, we will be in motion as we are compared with outside tree or building. Here, tree or builiding is a reference point.

When we are compared with the driver of the same bus, we are at rest. Here, we are not changing our position with respect to driver which is the refernce point. Therefore, rest and motion are relative terms because they depend upon the observer's reference point. 

Speed and Velocity

Speed:

Speed is defined as the distance travelled by a body per unit time in any direction. For example; when a body covers 10m distance in 1 sec, the speed of the body is 10m/s. It's SI unit is m/s. Speed has only magnitude but no direction. So, it is a scalar quantity (A physical quantity which has magnitude but no direction). Example; mass, length, time, speed, etc.

If a body travels a distance 's' in time 't', its speed 'v' will be given by:

Speed(v) = distance travelled(d) / time taken(t)

Velocity:

When a body moves from one point to another in a particular direction then its speed is called velocity. 
Velocity is defined as the distance travelled by a body in unit time in a particular direction.

OR

 Speed in a particular direction is called velocity.

For example; when a body covers 20 m distance in 1 sec towards east only then the velocity of the body is 20 m/s towards east.

Thus, velocity has magnitude as well as direction so it is a vector quantity (The physical quantity which has magnitude as well as a direction is called vector quantity). Example; displacement, force, velocity, acceleration, etc.

Velocity = distance travelled in a particular direction / time taken

Speed	Velocity
Rate of change of distance is called speed.	Rate of change of distance in a particular direction is called velocity.
It is a scalar quantity.	It is a vector quantity.
It's value is always positive.	It may be positive, negative, or zero.

Uniform motion, and non uniform or variable motion:

Uniform motion:

When a body covers equal distance in equal interval of time, its motion is called uniform motion. A body is said to have a uniform velocity if it travels equal distances in equal interval of time in a particular distance.

For example, a car covers 5m distance in each sec towards east only, so the car has uniform velocity or constant velocity 5m/s toeards east. (If a velocity of a body is equal throughout the motion, the body is said to be moving with uniform velocity or constant velocity.)

Non uniform motion:

When a body covers unequal distance in a equal interval of time, its motion is called non-uniform motion or variable motion. A body is said to have non-uniform velocity if it travels unequal distance in equal interval of time in a particular direction.

For example, car travels 5m in the first 1 sec, 10m in the second 1 sec, and 4m in the third 1 sec. Thus car travels different distance in equal interval of time and thus the motion of the car is variable motion or non uniform motion.

Average velocity:

When a body moves in a non-uniform velocity, its velocity changes continously. In such case, we have to calculate the average velocity.

Average velocity is defined as the ration of total distance covered by a body in a particular direction and the total time taken by the body.

Average velocity = Total distance travelled in a particular direction / total time taken = total displacement / time taken

In the above example, average velocity of the car = 5m + 10m + 4m / 1s +1s +1s = 19/3 m/s

Acceleration:

Acceleration is defined as the rate of change of velocity with time.

Acceleration = change in velocity / time taken

It's SI units is m/s². Suppose a body moves with initial velocity 'u', after time 't' its velocity changes into 'v'.

Here,

change in velocity = final velocity - initial velocity = v - u

acceleration = (v - u) / t

If the velocity of a moving body increases in every second, its acceleration will be positive. If the velocity of a moving body decreases in every second, its acceleration will be negative. The negative acceleration is called retardation or deceleration.

Uniform acceleration:

If the velocity of a moving body changes by equal amounts in equal intervals of time, then the body is said to be in uniform acceleration.

For example; the velocity of the car is increased by 1m/s in every second so the car has uniform acceleration 1m/s². When a body moves with uniform acceleration then its average velocity is given by half of sum of its initial velocity and final velocity. Avergae velocity = (u+v)/2 = 0+2/2 = 1m/s

Equation of motion:

If a body moves with uniform acceleration, its initial velocity 'u', final velocity 'v', distance travelled 's', time taken 't', and accelaration 'a' are related, the relation can be shown by equation. These equations of motion(suvat eqns):

Suppose a body moving with uniform accelaration has initial velocity 'u' after the time 't', the body covers the distance 's' and its final velocity reaches to 'v'. 

By the definition of acceleration, we have:

a = change in velocity / time taken = v-u / t

or, v-u = at

or, v= u + at ---(i)

For uniform acceleration, we have, 

average velocity = u+v / 2

Distance travelled = average velocity * time

s = (u+v /2) * t ---(ii)

Putting the value of v from the eqn (i) in eqn(ii), we have

s = (u+u+at / 2) * t

     = ut + 1/2at²

Putting the value of 't' from the eqn(i) in the eqn(ii), from eqn(i), t=v-u/a

we get, 

s= (u+v/2) (v-u/a)

or, v² - u² = 2a

Important notes for solving numerical problems:

1. If a body starts to move from rest, its initial velocity will be 0, u=0.
2. If a moving body comes to rest by applying brakes, its final velocity will be 0, v=0.
3. If a body moves with uniform velocity, the acceleraion a=0.
4. When a body is thrown vertically upwards, v=0, a = -g = -9.8 m/s²
5. When a body falls vertically downwards, u=0, a = +g = 9.8 m/s².

Solved Numerical Problems 

Q.  A bus is moving with initial velocity 10 m/sec, After 2 sec, the velocity becomes 20m/s. Find the acceleration of the bus.

Initial velocity (u)= 10m/s
Final velocity (v)=20m/s
Tere taken (t) = 2 sec
Acceleration (a) = ?

We know, 

a=v-u /t
a = (20m/s - 10m/s) /2 sec
a= 5 m/s²

Therefore the acceleration of the bus is 5 m/s².

Q. If a car covers a distance of 5.4km in 10 minutes, how much distance does it cover in 1 sec?

Here,

Distance travelled (s) = 5 4 km = 5.4 X 1000 = 5400m
Time taken(t) = 10 min = 10 *60 sec = 600 sec

Therefore, Distance travelled in 1 sec i.e. Speed (v) = s / t = 5400m / 600sec = 9 m/s

 

Q. A bus starts to move from rest and acquires a velocity of 30m/s in 10 seconds. Calculate its aceleration.

Here, 
initial velocity (u) = 0 (bus starts from rest)
final velocity (v) = 30m/s
time taken (t) =10s

We know, 

     a=( v-u)/t

or, a=(30-0)/10 m/s²

or, a = 3m/s²

 

Q. A car starts to move from rest. If the car gains an acceleeration of 3m/s² in 15 seconds. Calculate final velocity, and distance covered by the car.

Here,
u = 0
a = 3m/s²
t = 15s

Distance travelled (s) = ?

We have, 

     v = u + at

or, v =0+3m/s² * 15s

or, v = 45m/s

Again,

  s =(u+v /2) * t

or, s = (0 +45/2 * 15)m

or, s = 337.5m

Relative Velocity

The velocity of a body with respect to another body or a reference point is called relative velocity.

In the scenario of the Relative velocity of two bodies moving in the same direction:

Here, 

velocity of car A =Va

velocity of car B =Vb

Relative velocity of car A with respect to B (Vab)= Va - Vb

Relative velocity of car B with respect to A (Vba)= Vb - Va

In the scenario of the Relative velocity of two bodies moving in opposite direction:

Relative velocity of car A with respect to car B (Vab) = Va + Vb

Relative velocity of car B with respect to car A (Vba) = Vb + Va

Numericals:

Q. Two vehicles A and B are moving in the same direction with velocity 16m/s and 4m/s respectively. What is the velocity of first vehicle A with respect to B? What would be their velocity if they are moving in the opposite direction?

Given,

Velocity of vehicle A (Va) = 16m/s

Velocity of vehicle A (Vb) = 4m/s

In the context of they are moving in the same direction,

Relative velocity of car A with respect to B (Vab)= Va - Vb = (16 - 4)m/s = 12m/s

Relative velocity of car B with respect to A (Vba)= Vb - Va = (4 - 16)m/s = -12m/s

In the context of they are moving in the opposite direction,

Relative velocity of car A with respect to B (Vab)= Va + Vb = (16 + 4)m/s = 20m/s

Relative velocity of car B with respect to A (Vba)= Vb + Va = (4 + 16)m/s = 20m/s

Q. Two cars A and B are moving with velocities 30m/s and 15m/s respectively in the opposite direction. If they started from the same place at the same time. What would be the distance between them in 2 minutes?

Given,

Velocity of car (Va) = 30m/s

Velocity of car (Vb) = 15m/s

Time (t) = 2 min = 120 seconds

Distance covered by the car A in 2 minutes (s1) =  Va * t = 30 * 120 = 3600m

Distance covered by the car B in 2 minutes (s2) =  Vb * t = 15 * 120 = 1800m

Distance between them after two minutes will be = s1 - s2 = (3600 - 1800)m = 1800m