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 #?

1 Answer
Feb 15, 2016

Answer:

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.