The velocity of an object with a mass of $4 k g$ $4 kg$ is given by $v (t) = sin 3 t + cos 7 t$ $v(t)= sin 3 t + cos 7 t$ . What is the impulse applied to the object at $t = \frac{5 π}{12}$ $t= (5pi)/12$ ?

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The velocity of an object with a mass of $4 k g$ $4 kg$ is given by $v (t) = sin 3 t + cos 7 t$ $v(t)= sin 3 t + cos 7 t$ . What is the impulse applied to the object at $t = \frac{5 π}{12}$ $t= (5pi)/12$ ?

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Answer 1 · 2016-06-04T22:04:26

It is $- 6.69$ $-6.69$ $k g \frac{m}{s}$ $kgm/s$ .

Explanation:

In classical mechanics and with constant mass, the instantaneous impulse is defined as the momentum

$p = m v$ $p=mv$

In your case

$p = 4 \cdot [sin (3 t) + cos (7 t)]$ $p=4*[sin(3t)+cos(7t)]$

and this has to be evaluated when $t = \frac{5 π}{12}$ $t=(5pi)/12$

then

$p = 4 \cdot [sin (3 \cdot \frac{5 π}{12}) + cos (7 \cdot \frac{5 π}{12})]$ $p=4*[sin(3*(5pi)/12)+cos(7*(5pi)/12)]$

$= 4 \cdot [sin (\frac{5}{4} π) + cos (\frac{35}{12} π)]$ $=4*[sin(5/4pi)+cos(35/12pi)]$

we have that $sin (\frac{5}{4} π) = - sin (\frac{π}{4})$ $sin(5/4pi)=-sin(pi/4)$ because the angle is $π + \frac{π}{4}$ $pi+pi/4$ and $cos (\frac{35}{12} π) = cos (\frac{11}{12} π)$ $cos(35/12pi)=cos(11/12pi)$ because the angle is $2 π + \frac{11}{12} π$ $2pi+11/12pi$ .
So we have

$p = 4 \cdot [- sin (\frac{π}{4}) + cos (\frac{11}{12} π)]$ $p=4*[-sin(pi/4)+cos(11/12pi)]$

$= 4 \cdot (- \frac{1}{\sqrt{2}} - \frac{1 + \sqrt{3}}{2 \sqrt{2}})$ $=4*(-1/sqrt(2)-(1+sqrt(3))/(2sqrt(2)))$

$= 4 \cdot (- \frac{3 + \sqrt{3}}{2 \sqrt{2}})$ $=4*(-(3+sqrt(3))/(2sqrt(2)))$

$= - \frac{6 + 2 \sqrt{3}}{\sqrt{2}}$ $=-(6+2sqrt(3))/(sqrt(2))$

$= - \frac{6 \sqrt{2} + 2 \sqrt{6}}{2}$ $=-(6sqrt(2)+2sqrt(6))/2$

$= - 3 \sqrt{2} - \sqrt{6} = - \sqrt{2} (3 + \sqrt{3}) \approx - 6.69$ $=-3sqrt(2)-sqrt(6)=-sqrt(2)(3+sqrt(3))\approx-6.69$ $k g \frac{m}{s}$ $kgm/s$ .

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The velocity of an object with a mass of $4 k g$ $4 kg$ is given by $v (t) = sin 3 t + cos 7 t$ $v(t)= sin 3 t + cos 7 t$ . What is the impulse applied to the object at $t = \frac{5 π}{12}$ $t= (5pi)/12$ ?

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The velocity of an object with a mass of 4kg4 kg is given by v(t)=sin3t+cos7tv(t)= sin 3 t + cos 7 t . What is the impulse applied to the object at t=5π12t= (5pi)/12 ?

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Explanation:

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The velocity of an object with a mass of $4 k g$ $4 kg$ is given by $v (t) = sin 3 t + cos 7 t$ $v(t)= sin 3 t + cos 7 t$ . What is the impulse applied to the object at $t = \frac{5 π}{12}$ $t= (5pi)/12$ ?