Question

The velocity of an object with a mass of `8 kg` is given by `v(t)= sin 3 t+ cos 2 t`. What is the impulse applied to the object at `t= ( 3 pi)/ 4`?

Answer 1

Please see the explanation ...

Explanation:

This is an ill-posed problem. I see a whole lot of questions asking What is the impulse applied to an object at a given instant. You can talk about force applied at a given instant. But when we talk about Impulse, it is always defined for a time interval and not for an instant of time.

By Newton's Second Law,
Force: `\vec{F}=\frac{d\vec{p}}{dt}=\frac{d}{dt}(m.\vec{v})=m\frac{d\vec{v}}{dt}`

Magnitude of the force : `F(t)=m\frac{dv}{dt}=m.\frac{d}{dt}(sin3t+cos2t)`,
`F(t)= m.(3cos3t-2sin2t)`

`F(t=(3\pi)/4)=(8 kg)\times(3cos((9\pi)/4)-2sin((3\pi)/2))ms^{-2}=32.97 N`

Impulse : `J=\int_{t_i}^{t_f} F(t).dt` is defined for the time interval `\Delta t=t_f-t_i`. So it makes no sense to talk about impulse at an instant.