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

Jun 4, 2016

From the basic theory of dynamics, if $v \left(t\right)$ is the velocity and $m$ be the mass of an object, $p \left(t\right) = m v \left(t\right)$ is it's momentum.

Another result of Newton's second law is that, Change in momentum = Impulse

#### Explanation:

Assuming that the particle moves with the constant velocity $v \left(t\right) = S \in 4 t + C o s 4 t$ and a force acts on it to stop it completely, we shall calculate the impulse of the force on the mass.

Now the momentum of the mass at $t = \frac{\pi}{4}$ is,

${p}_{i} = 3 \left(S \in 4 \cdot \frac{\pi}{4} + C o s 4 \cdot \frac{\pi}{4}\right) = 3 \left(S \in \pi + C o s \pi\right) = - 3$ units.

If the body/particle is stopped the final momentum is $0$.

Thus, ${p}_{i} - {p}_{f} = - 3 - 0$ units.

This is equal to the impulse of the force.

Thus, $J = - 3$ units.

The negative sign arises because the external force and hence it's impulse acts opposite to the particle's motion. If the particle's motion is assumed to be in the positive direction, the impulse is in the negative direction.

We have also assumed that the force stops the particle at the instant $t = \frac{\pi}{4}$.

I hope it helped.