Question

A car starts from the rest and moves with a uniform acceleration ‘a’ on a straight road from time t = 0 to t = T. A constant deceleration brings it a rest. In this process the average speed of the car is
A) $\dfrac{{aT}}{4}$
B) $\dfrac{{3aT}}{2}$
C) $\dfrac{{aT}}{2}$
D) $aT$

Answer 1

Hint
Here we use the Newton first equation of motion to find out the value maximum velocity attained by the car i.e. $v= u+at$.
As we also need to find out distance travelled by the car so we use the formula. Newton's second law of motion and to find the average speed of the car we use the formula-
Avg speed = $\dfrac{s}{t}$.

Complete step by step answer
Dividing it into 2 parts where first part is acceleration and second part is the deceleration. As we have to find the average speed so firstly we find the maximum speed of the car
$v_{max} = at$.
Where, ‘$a$’ is the acceleration,
‘$t$‘ is the time.
Then we will calculate time taken to stop the car. So for this we use the formula-
$v= u+at$;
As final velocity is zero and the initial velocity during retardation will be the $v_{max}$ because from that moment the car starts retardation. So now put the value. The car retardation will denote it by “$r$”
$0 = at – rt$
So, Acceleration is equal to retardation and to find out the distance travelled by the car during retardation we use the formula Newton's second law of motion-
$v^2 – u^2 = 2as$.
As during retardation final velocity is equal to zero then on putting value in the equation-
$s = \dfrac{{\;at}}{2}$ $\dfrac{{\;{u^2}}}{{2r}}$
Where (r = -a)
So, $S = \dfrac{{\;{{\left( {at} \right)}^2}}}{{2r}}$.
So, finally putting the formula of average speed
Avg speed = $\dfrac{s}{t}$
Avg speed = $\dfrac{{\;at}}{2}$
Option (C) is the correct answer.

Note
We have to always remember that Newton equations of motion can always be applied when the body possesses uniform or constant acceleration.