Question

The speedometer readings of a car are shown below. Find the acceleration of the car and its displacement.

Time	Speedometer
9:25 am	36 km/h
9:45 am	72 km/h

Answer 1

Hint:Assume that car is travelling with uniform acceleration. Use the formula for acceleration of a body moving with constant acceleration. Then use a suitable kinematic equation to find the displacement of the car.

Formula used:
$a=\dfrac{v-u}{t}$
$2as={{v}^{2}}-{{u}^{2}}$,
where $a$ is constant or average acceleration, $v$ is final velocity, $u$ is initial velocity, $s$ is displacement and $t$ is time.

Complete step by step answer:
Let us assume that the car is moving along a straight line in a single direction and at a constant acceleration. Then the speed of the car is equal to the velocity of the car. The acceleration of a body is defined as the rate of change in velocity of the body with respect to time. If the body is moving with constant acceleration, then its acceleration is equal to
$a=\dfrac{v-u}{t}$ …. (i)
Here, $u=36km{{h}^{-1}}=36\times \dfrac{{{10}^{3}}m}{3600s}=10m{{s}^{-1}}$.
And $v=72km{{h}^{-1}}=72\times \dfrac{{{10}^{3}}m}{3600s}=20m{{s}^{-1}}$.
From the table we get that the car is moving with velocity $10m{{s}^{-1}}$ at 9:25 am and with velocity $20m{{s}^{-1}}$ at 9:45 am. This means that the time taken to increase the velocity from u to v is $t=20min=20\times 60=1200s$.Substitute these values in (i).
$a=\dfrac{20-10}{1200}\\
\Rightarrow a=\dfrac{10}{1200}\\
\Rightarrow a=\dfrac{1}{120}m{{s}^{-2}}$
To find the displacement of the car, we shall use the kinematic equation $2as={{v}^{2}}-{{u}^{2}}$ …. (ii).
Substitute the values of a, v and u in (ii).
$\Rightarrow 2\left( \dfrac{1}{120} \right)s={{(20)}^{2}}-{{(10)}^{2}}$
$\Rightarrow \left( \dfrac{1}{60} \right)s=400-100$
$\therefore s=60\times 300=18000m=18km$

Therefore, the acceleration of the car is $\dfrac{1}{120}m{{s}^{-2}}$ and its displacement for that time is 18km.

Note: The kinematic equations that we have used are only valid in the case when the acceleration of the body is constant. Therefore, we had to assume that the acceleration of the car is constant. Also note that the speedometer of a car shows the speed of the car and not the velocity. However, if the car is travelling in a single direction then we consider the speed as velocity.