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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 #?

Answer 1

Charles Aeschliman

There is no impulse at that time.

The acceleration, #a(t)#, is the first derivative of #v(t)#.

#a(t) = (dv(t))/dt = 2cos2t -4sin4t#

Evaluating that at time #t = pi/4#,

#a(pi/4) = 2cos(pi/2) - 4sinpi = 2*0 - 4*0 = 0#

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

Now impulse. Impulse is #"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 = 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

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Answer 2

Emily Abbate

The impulse applied to the object at ( t = \frac{\pi}{4} ) is ( (-2\sqrt{2}, -4\sqrt{2}) ) N·s.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.