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**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

###### FREE Expert Solution

**(a)** Center of mass velocity, v_{cm} = dx_{cm}/dt

x_{cm} = (m_{1}x_{1} + m_{2}x_{2})/(m_{1} + m_{2})

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

###### Problem Details

The blocks shown can only move along the *x*-axis.**(a)** Their velocities at a certain moment are *v*_{1} and *v*_{2} . Find the velocity of the center of mass *v _{cm}* at that moment. Keep in mind that, in general:

*v*

*. Give your answer in terms of*

_{x}= dx/dt*m*

_{1},

*m*

_{2},

*v*

_{1}, and

*v*

_{2}.

**(b)**The blocks' momenta at a certain moment are P

_{1}and P

_{2}. Find the

*x*component of the velocity of the center of mass

*v*at that moment. Give your answer in terms of

_{cm}*m*

_{1},

*m*

_{2}, P

_{1}, and P

_{2}.

**(c)**The blocks' accelerations at a certain moment are

*a*

_{1}and

*a*

_{2}. Find the acceleration of the center of mass

*a*

_{cm}

_{ }at that moment. Keep in mind that, in general,

*a*

*=*

_{x }*d*

*v*

*/*

_{x}*d*

*t*. Give your answer in terms of

*m*

_{1},

*m*

_{2},

*a*

_{1}, and

*a*

_{2}.

Frequently Asked Questions

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.