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Question

The position x of a particle varies with time t as x=αt2βt3. Which of the following is (are) correct?

A
The particle will return to its starting point after time αβ.
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B
The particle will come to rest after time 2α3β.
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C
The initial velocity of the particle was zero but its initial acceleration was not zero.
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D
The acceleration of the particle is zero at t=α3β.
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Solution

The correct options are
A The particle will return to its starting point after time αβ.
B The particle will come to rest after time 2α3β.
C The initial velocity of the particle was zero but its initial acceleration was not zero.
D The acceleration of the particle is zero at t=α3β.
Differentiate x=αt2βt3 w.r.t time to get velocity
v=dxdt=2αt3βt2.....(1)
Differentiate v w.r.t time to get acceleration
a=dvdt=2α6βt....(2)
The particle will return to its initial position when x=0
i.e., 0=αt2βt3
which gives t=αβ
The particle will come to rest when v=0
i.e., 0=2αt3βt2
which gives t=2α3β
Using equation (1) and (2), we say that initial velocity of the particle is zero but initial acceleration is a=2α
a=02α6βt=0, which gives t=α3β
Thus, acceleration is zero at t=α3β

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