Question

The acceleration of a motorcycle is given as a function of time ${a_x}(t) = At - B{t^2}$ , where $A = 1.50\,m\,{\sec ^{ - 2}}$ and $B = 0.120\,m\,{\sec ^{ - 4}}$ . The motorcycle is at rest at origin at $t = 0$. Calculate the maximum velocity that the motorcycle attains:

Answer 1

Hint: In order to find the maximum velocity we will first find the time at which acceleration is zero and since acceleration is a derivative of velocity of a body with respect to time, using this concept we will solve for velocity at particular time. We will use the relation between velocity and acceleration which is $a = \dfrac{{dv}}{{dt}}$.

Complete step by step answer:
First, it’s given us that at $t = 0$ motorcycle was at rest at origin which means the value of $t = 0$ since, body is at origin. For the motorcycle to have a maximum velocity its acceleration must be zero and we will put acceleration magnitude zero in the given relation ${a_x}(t) = At - B{t^2}$ and we will get,
$At - B = 0$
$\Rightarrow t = \dfrac{A}{B}$
Putting the values $A = 1.50\,m\,{\sec ^{ - 2}}$ and $B = 0.120\,m\,{\sec ^{ - 4}}$ we get,
$t = 12.5\sec $
Now, acceleration can be written as:
$\dfrac{{dv}}{{dt}} = At - B{t^2}$
Or integrating both sides,
$\int {dv} = \int {(At - B{t^2})dt} $
$\Rightarrow v = A\dfrac{{{t^2}}}{2} + B\dfrac{{{t^3}}}{3}$
On putting the values of
$t = 12.5\sec $
$\Rightarrow A = 1.50\,m\,{\sec ^{ - 2}}$
$\Rightarrow B = 0.120\,m\,{\sec ^{ - 4}}$
In the equation $v = A\dfrac{{{t^2}}}{2} - B\dfrac{{{t^3}}}{3}$
We get,
$v = (0.75){(12.5)^2} - (0.04){(12.5)^3}$
$\therefore v = 39.1\,m\,{\sec ^{ - 1}}$
So when acceleration is zero the magnitude of the velocity of the motorcycle is maximum.

Hence, the value of maximum velocity of the motorcycle is $v = 39.1\,m{\sec ^{ - 1}}$.

Note: It should be remembered that, the basic integration formula of functions like $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} $ and since motorcycle was at origin hence, the definite integral can be ignored since it was then when $t = 0$ and just at this time motorcycle attains its maximum velocity and also remember the relation between velocity and acceleration which is $a = \dfrac{{dv}}{{dt}}$.