# The velocity of an object with a mass of #6 kg# is given by #v(t)= te^t-t^2 #. What is the impulse applied to the object at #t= 5 #?

The question is a trick question. Impulse is defined over a time interval. Impulse at one single instant is zero by definition for any finite force acting on the object.

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To find the impulse applied to the object at ( t = 5 ), you need to calculate the change in momentum of the object over a small time interval around ( t = 5 ).

Impulse = Change in momentum

Change in momentum = Final momentum - Initial momentum

To find the final momentum at ( t = 5 ), you first need to find the velocity of the object at ( t = 5 ) using the given velocity function. Then, multiply the velocity by the mass of the object.

Initial momentum is zero, as the object starts from rest at ( t = 0 ).

Given velocity function: ( v(t) = te^t - t^2 )

At ( t = 5 ): ( v(5) = (5)e^5 - (5)^2 )

Now, calculate the final momentum at ( t = 5 ): ( p_f = m \cdot v(5) )

Where ( m = 6 ) kg.

Finally, the impulse applied to the object at ( t = 5 ) is the change in momentum: ( \text{Impulse} = p_f - p_i )

Where ( p_i = 0 ) (initial momentum).

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To find the impulse applied to the object at ( t = 5 ), we need to calculate the change in momentum of the object over the interval ( [0, 5] ).

Impulse is defined as the change in momentum, which can be expressed as the integral of force with respect to time. In this case, the force acting on the object is given by the derivative of velocity with respect to time, ( \frac{dv}{dt} ).

So, the impulse can be calculated using the integral of the force function ( \frac{dv}{dt} ) over the time interval ( [0, 5] ):

[ \text{Impulse} = \int_{0}^{5} \frac{dv}{dt} , dt ]

Given that the velocity function is ( v(t) = te^t - t^2 ), we need to find the derivative of ( v(t) ) with respect to ( t ) to obtain the force function ( \frac{dv}{dt} ).

[ \frac{dv}{dt} = \frac{d}{dt}(te^t - t^2) ]

[ = e^t + te^t - 2t ]

Now, we can evaluate this expression at ( t = 5 ) and integrate it over the interval ( [0, 5] ) to find the impulse applied to the object at ( t = 5 ).

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