Question

A particle starts from rest at time \[t = 0\] and moves on a straight line on an acceleration which varies as shown in fig. The speed of the particle will be maximum after how many seconds?

A. $4$
B. $5$
C. $6$
D. $1$

Answer 1

Hint:Use the first kinematic equation. This kinematic equation gives the relation between the final velocity of particle, initial velocity of particle, acceleration of particle and time. Determine the velocity of the particle after 2 seconds, 4 seconds and 6 seconds. Check at which time the speed of the particle is maximum to determine the final answer.

Formula used:
The kinematic equation for final velocity \[v\] of an object is
\[v = u + at\] …… (1)
Here, \[u\] is initial velocity of the object, \[a\] is acceleration of the object and \[t\] is time.

Complete step by step answer:
We have given that the particle starts from rest. Hence, the initial velocity of the particle is zero.
\[u = 0\,{\text{m/s}}\]
Let us consider the travel of the particle from time \[t = 0\,{\text{s}}\] to \[t = 2\,{\text{s}}\]. Let us first determine the velocity of the particle at time \[t = 2\,{\text{s}}\].
At time \[t = 0\,{\text{s}}\], the acceleration of the particle is.
\[a = - 10\,{\text{m/}}{{\text{s}}^2}\]
Substitute \[0\,{\text{m/s}}\] for \[u\], \[ - 10\,{\text{m/}}{{\text{s}}^2}\] for \[a\] and \[2\,{\text{s}}\] for \[t\] in equation (1).
\[v = \left( {0\,{\text{m/s}}} \right) + \left( { - 10\,{\text{m/}}{{\text{s}}^2}} \right)\left( {2\,{\text{s}}} \right)\]
\[ \Rightarrow v = - 20\,{\text{m/s}}\]
Hence, the velocity of the particle after 2 seconds is \[ - 20\,{\text{m/s}}\].

Let us consider the travel of the particle from time \[t = 2\,{\text{s}}\] to \[t = 6\,{\text{s}}\].
Let us first determine the velocity of the particle at time \[t = 6\,{\text{s}}\].
At time \[t = 2\,{\text{s}}\], the acceleration of the particle is.
\[a = 10\,{\text{m/}}{{\text{s}}^2}\]
Substitute \[ - 20\,{\text{m/s}}\] for \[u\], \[10\,{\text{m/}}{{\text{s}}^2}\] for \[a\] and \[6\,{\text{s}}\] for \[t\] in equation (1).
\[ \Rightarrow v = \left( { - 20\,{\text{m/s}}} \right) + \left( {10\,{\text{m/}}{{\text{s}}^2}} \right)\left( {6\,{\text{s}}} \right)\]
\[ \therefore v = 40\,{\text{m/s}}\]
Hence, the velocity of the particle after 4 seconds is \[40\,{\text{m/s}}\] which is the maximum speed of the particle. Therefore, the speed of particles is maximum after 6 seconds.

Hence, the correct option is C.

Note:The students should remember that we have asked the maximum speed of the particle and not the maximum velocity. Hence, if the value of velocity after 6 seconds is negative of the obtained value then also the maximum speed will be after 6 seconds as the speed is magnitude of velocity of the particle. Also, the velocity of the particle after 4 seconds is zero, hence, it is not calculated.