A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $$m=0.4kg$$ is at rest on this surface. An impulse of $$1.0Ns$$ is applied to the block at time $$t=0$$ so that it starts moving along the $$x-axis$$ with a velocity $$vt=v0e$$−$$t/$$τ, where $$v0$$ is a constant and τ$$=4s$$. The displacement of the block, in metres, at $$t=$$τ is Take e−$$1=0.37$$.

Question

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $$m=0.4kg$$ is at rest on this surface. An impulse of  $$1.0Ns$$ is applied to the block at time  $$t=0$$ so that it starts moving along the $$x-axis$$ with a velocity  $$vt=v0e$$−$$t/$$τ, where $$v0$$  is a constant and  τ$$=4s$$. The displacement of the block, in metres, at  $$t=$$τ  is Take  e−$$1=0.37$$.

Infinity Learn · Accepted Answer

$$V=V0e$$−$$t/$$τ                    here  $$V0=Jm=2.5m/s$$ ⇒$$dxdt=V0e$$−$$t/$$τ⇒∫$$0xdx=V0$$∫$$0$$τe−$$t/$$τdt           Since∫e−$$xdx=e$$−x−$$1$$⇒$$x=V0e-t/$$τ$$-1$$τ⇒$$x=2.5$$ −$$4$$ e−$$1$$−$$e0$$                  ​⇒$$x=2.5$$−$$4$$ $$0.37$$−$$1$$​⇒$$x=6.30$$

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