Motion of objects we see around us are complex in nature. However, we take just one type of motion called linear motion and arrive at three equations of linear motion by Mathematical modelling. We also will solve some problems by using the equations of linear motion.

**Distance and displacement**

Let a car travel from A to B, B to C , C to D and back to C on a straight highway. The total path length covered by the car is AB+BC+2CD. This is the distance travelled by the car.

The displacement of the car is AC which is the shortest distance from the starting position to the final position of the car.

Let the positions B, C and D be at distances 20 km, 50 km and 90 km from the starting position A. So the total distance travelled by the car when it travels from A to B and back to C is 90+40=130 km. Whereas the displacement is only 50 km!

If the car travels back to A, then the total distance travelled will be 180 km whereas its displacement will be 0 km!

Let us solve a simple problem related to distance and displacement.

A farmer goes from the point B to C and then to D along the boundary of his rectangular field as shown in the figure. What is (i) distance travelled by him ? (ii) his displacement?

Solution: distance=70 m

displacement= 50 m

**Uniform motion**

A body has uniform motion if it covers equal distances in equal intervals of time.

**Average speed**

The average speed of an object is calculated by dividing the total distance travelled by the total time taken.

e.g. An object travels 20m in 5 seconds and then another 20 m in 3 seconds. Find its average speed.

Solution: Total distance travelled= 20m+20m=40m

Total time taken = 5s+3s =8s

**Velocity**

Velocity of a body is its speed in a given direction.

Velocity of a moving body changes when there is a change in either its speed or in direction or in both.

**Average Velocity**

When a body, moving in a straight line, changes its velocity at uniform rate, then its average velocity is given by the arithmetic mean of initial velocity and final velocity.

The rate of change of velocity of a body moving in a straight line is called its accelerations.

where u, v, a and t are initial velocity, final velocity, acceleration and time respectively.

**Uniform acceleration**

A body has uniform acceleration if it changes the velocity at uniform rate. That is, the change in velocity at equal intervals of time remain the same.

**Equations of linear motion**

Following are the equations of linear motion.

Now let us see how we got these equations of linear motion.

(i) **Equation for velocity – time relation**

Consider a body moving in a straight line with uniform acceleration a. If it changes its velocity from u to v in time t, then by the definition of acceleration, we have

On simplification, we get at = v-u

On rearrangement, we get v = u + at

(ii) **Equation for position – time relation**

Consider a body moving in a straight line with uniform acceleration a. If it changes its velocity from u to v in time t after covering a distance s, we have

(iii)** Equation for position – velocity relation.**

Consider a body moving in a straight line with uniform acceleration a. If it changes its velocity from u to v in time t after covering a distance s, we have

Are you ready for solving few problems based on the equations of linear motion?

1. A train travelling with a speed of 36km/h is brought to rest in 10 seconds by applying brake. If it retards at uniform rate, find the distance travelled before it stops.

Hope you enjoyed the discussion.

Are you ready for a quiz?

Answer the following questions

1. A ball is thrown vertically up by a person and it raised to a height of 10m and caught by him when it descended.What is the distance travelled by the ball? What is its displacement?

2. A stone is dropped from a certain height. How much distance will it fall if its acceleration is 9.8 ms^-2

3. A train starting from rest attains a velocity of 18 km/h in 5 seconds. What is its acceleration?

4. A train moving with uniform acceleration changes its speed from 18 km/h to 36 km/h in 10 seconds. What is its acceleration? How much distance will it travel during this time?

5. A train travelling with a uniform speed of 72 km/h clears a post in 10 seconds. What is the length of the train?

Answers:

1. 20m, 0 m

2. 9.9m

3. 1 ms^-2

4. a = 1/2 ms^-2, s= 75 m

5. 200m