A classical mechanics problem by A Former Brilliant Member

Classical Mechanics Level 2

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m = 0.4$ kg is rest on its surface. An impulse of $1.0Ns$ is applied to the block at time $t = 0$ and it starts moving along X-direction with velocity $v_t = v_0 e^-t/T$ ,where $v_0$ is a constant and $T= 4s.$ The displacement of block , in meters at $t = T$ is

The answer is 6.3.

1 solution

A Former Brilliant Member
Nov 25, 2018

We know that, Change in momentum = Impulse

$F\times t = mv_0 \Rightarrow 1.0 = 0.4\times v_0 \Rightarrow v_0 = 2.5 ms^{-1}$

We also know that , $\dfrac {ds}{dt} = v_t = v_0e^{-t/T}= 2.5e^{-t/T}$ , where $s$ is the displacement.

$\Rightarrow \int_0^s ds = \int_0^t 2.5 e^{-t/T} dt$

$\Rightarrow s = 2.5\times \dfrac {[e^{-t/T}]_0^t}{-\dfrac 1T}$

At $t = T = 4s$

$s = 2.5\times 4 [e^0 - e^{-1}] = 10\times 0.63= \boxed{6.3m}$

I am confused by the way you found the initial velocity. The question states the impulse is applied at t=0 so wouldn't Ft at t=0 imply zero initial velocity and require the impulse to have acted for one second to get the initial velocity you calculated?

Tristan Goodman - 2 years, 6 months ago

That $v_0$ is the velocity aquired at the instant the Impulse is acted and see that I have given the relation of velocity with time and according to that relation for $v_0$ to be zero you need $e^{t/T}$ to be zero which is not possible at $t=0.$

And its a oil covered surface, which is other way of saying its frictionless surface, so no force is required to overcome friction hence the block aquires velocity at the instant Impulse is acted on it.

A Former Brilliant Member - 2 years, 6 months ago

A classical mechanics problem by A Former Brilliant Member

The answer is 6.3.

1 solution

0 pending reports