Question

A car accelerates from rest at a constant rate \[\alpha \] for some time after which it decelerates at a constant rate \[\beta \] to come to rest. If the total time elapsed is t, the maximum velocity acquired by the car is given by
A. \[\left( \dfrac{{{\alpha }^{2}}+{{\beta }^{2}}}{\alpha \beta } \right)t\]
B. \[\left( \dfrac{{{\alpha }^{2}}-{{\beta }^{2}}}{\alpha \beta } \right)t\]
C. \[\left( \dfrac{\alpha +\beta }{\alpha \beta } \right)t\]
D. \[\left( \dfrac{\alpha \beta }{\alpha +\beta } \right)t\]

Answer 1

Hint: In this question we have been asked to calculate the maximum velocity acquired by the car during acceleration and deceleration. We have been given the acceleration and deceleration of the car along with the time the car takes to come to rest. Therefore, to solve this question, we shall use the equation of kinetic motion. We shall first calculate the velocity during acceleration and next during deceleration. We shall sum up the time required for both the process as it is given as t.

Formula Used: \[v=u+at\]

Complete answer:
It is given that a car accelerates from rest with constant rate \[\alpha \]. Now, let the car accelerate for time \[{{t}_{1}}\] before decelerating.
Therefore, using second equation of motion
We get,
\[v=0+\alpha {{t}_{1}}\]
On solving,
\[{{t}_{1}}=\dfrac{v}{\alpha }\] …………… (1)
Now, similarly let \[{{t}_{2}}\]be the time it takes to come to rest after decelerating
Therefore,
\[0=v-\beta {{t}_{2}}\]
On solving,
\[{{t}_{2}}=\dfrac{v}{\beta }\] …………….. (2)
Now, we have been given that total time taken by the car to come to rest is t
Therefore,
\[t={{t}_{1}}+{{t}_{2}}\] …………… (3)
From (1), (2) and (3)
We get,
\[t=\dfrac{v}{\alpha }+\dfrac{v}{\beta }\]
On solving,
\[v=\left( \dfrac{\alpha \beta }{\alpha +\beta } \right)t\]

Therefore, the correct answer is option D.

Note:
The equations of motion are a set of equations that are used to describe the motion of a particle or system. These equations relate the parameters such as initial and final velocity, acceleration, distance and time. The equations of motion are applicable to any system having constant acceleration or deceleration.