The kinetic energy of an object with a mass of $2 kg$ constantly changes from $8 J$ to $136 J$ over $4 s$ . What is the impulse on the object at $1 s$ ?

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Answer 1 · 2016-08-20T13:06:35

$vec J_(0 to 1) = 4(sqrt( 10) - sqrt( 2) )hat p$ N s

I think there is something wrong in the formulation of this question.

With Impulse defined as
$vec J = int_(t = a)^b vec F(t) \ dt$
$= int_(t = a)^b vec dot p (t) \ dt = vec p(b) - vec p(a)$

then the Impulse on the object at t= 1 is

$vec J = int_(t = 1)^1 vec F(t) \ dt = vec p(1) - vec p(1) = 0$

It may be that you want the total impulse applied for $t in [0,1]$ which is

$vec J = int_(t = 0)^1 vec F(t) \ dt = vec p(1) - vec p(0) qquad star$

To evaluate $star$ that we note that if the rate of change of kinetic energy $T$ is constant, ie:

$(dT)/(dt) = const$

then

$T= alpha t + beta$

$T(0) = 8 implies beta = 8$

$T(4) = 136 = alpha(4) + 8 implies alpha = 32$

$T= 32 t + 8$

Now $T = abs(vec p)^2/(2m)$ .

$implies (vec p * vec p)= 4(32 t + 8)$

$vec p = 2sqrt( (32 t + 8)) hat p$

and
$vec p(1) - vec p (0)$

$= (2sqrt( (32 + 8)) - 2sqrt( 8) )hat p$

$= 4(sqrt( 10) - sqrt( 2) )hat p$ N s

The kinetic energy of an object with a mass of 2 kg2 kg constantly changes from 8 J8 J to 136 J136 J over 4 s4 s. What is the impulse on the object at 1 s1 s?