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

May 18, 2018

There is no impulse at that time.

Explanation:

The acceleration, $a \left(t\right)$, is the first derivative of $v \left(t\right)$.

$a \left(t\right) = \frac{\mathrm{dv} \left(t\right)}{\mathrm{dt}} = 2 \cos 2 t - 4 \sin 4 t$

Evaluating that at time $t = \frac{\pi}{4}$,

$a \left(\frac{\pi}{4}\right) = 2 \cos \left(\frac{\pi}{2}\right) - 4 \sin \pi = 2 \cdot 0 - 4 \cdot 0 = 0$

Since the acceleration is zero, the force applied to this object is zero.

Now impulse. Impulse is $\text{force"*"time}$, where time is the length of time that the force was applied. The question does not refer to a time duration, just the moment when $t = \frac{\pi}{4}$.

So "impulse" would not really apply to this situation even if the force had not calculated out to be zero.

I hope this helps,
Steve