# Problem: The blocks shown can only move along the  x-axis.(a) Their velocities at a certain moment are  v1  and  v2 . Find the velocity of the center of mass  vcm  at that moment. Keep in mind that, in general:  vx = dx/dt. Give your answer in terms of  m1,  m2,  v1, and  v2.(b) The blocks' momenta at a certain moment are P1 and P2 . Find the  x  component of the velocity of the center of mass  vcm  at that moment. Give your answer in terms of  m1,  m2, P1, and P2.(c) The blocks' accelerations at a certain moment are  a1  and  a2 . Find the acceleration of the center of mass  acm  at that moment. Keep in mind that, in general,  ax = dvx/dt. Give your answer in terms of  m1,  m2,  a1, and  a2.

###### FREE Expert Solution
79% (352 ratings)
###### FREE Expert Solution

(a) Center of mass velocity, vcm = dxcm/dt

xcm = (m1x1 + m2x2)/(m1 + m2)

$\begin{array}{rcl}{\mathbf{v}}_{\mathbf{cm}}& \mathbf{=}& \frac{\mathbf{d}}{\mathbf{dt}}\mathbf{\left(}\frac{{\mathbf{m}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}}{{\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}}\mathbf{\right)}\\ & \mathbf{=}& \mathbf{\left(}\frac{\mathbf{1}}{{\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}}\mathbf{\right)}\frac{\mathbf{d}}{\mathbf{dt}}\mathbf{\left(}{\mathbf{m}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}\\ & \mathbf{=}& \mathbf{\left(}\frac{\mathbf{1}}{{\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}}\mathbf{\right)}\mathbf{\left(}{\mathbf{m}}_{\mathbf{1}}{\mathbf{v}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{v}}_{\mathbf{2}}\mathbf{\right)}\end{array}$

79% (352 ratings)
###### Problem Details

The blocks shown can only move along the  x-axis.
(a) Their velocities at a certain moment are  v1  and  v2 . Find the velocity of the center of mass  vcm  at that moment. Keep in mind that, in general:  vx = dx/dt. Give your answer in terms of  m1,  m2,  v1, and  v2.
(b) The blocks' momenta at a certain moment are P1 and P2 . Find the  x  component of the velocity of the center of mass  vcm  at that moment. Give your answer in terms of  m1,  m2, P1, and P2.
(c) The blocks' accelerations at a certain moment are  a1  and  a2 . Find the acceleration of the center of mass  acm  at that moment. Keep in mind that, in general,  a= dvx/dt. Give your answer in terms of  m1,  m2,  a1, and  a2.

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What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Intro to Center of Mass concept. You can view video lessons to learn Intro to Center of Mass Or if you need more Intro to Center of Mass practice, you can also practice Intro to Center of Mass practice problems .

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Based on our data, we think this problem is relevant for Professor Xiong's class at PITT.