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Chapter 7: Problem 35

Body A of mass \(3 \mathrm{kg}\) is moving in the \(+x\)-direction with a speed of \(14 \mathrm{m} / \mathrm{s}\). Body \(\mathrm{B}\) of mass \(4 \mathrm{kg}\) is moving in the-y-direction with a speed of \(7 \mathrm{m} / \mathrm{s}\). What are the \(x\) - and \(y\) -components of the velocity of the \(\mathrm{CM}\) of the two bodies?

Short Answer

Expert verified
Answer: The x-component of the velocity of the center of mass is 6 m/s, and the y-component is -4 m/s.

Step by step solution

01

Identify given values

We are given the following information: - Body A: mass \(m_A = 3 \ kg\), velocity \(v_A = 14 \ m/s\) (in the +x-direction) - Body B: mass \(m_B = 4 \ kg\), velocity \(v_B = 7 \ m/s\) (in the -y-direction)
02

Calculate the total mass

To find the center of mass velocity, first calculate the total mass of the two bodies: $$ M = m_A + m_B = 3 + 4 = 7 \ kg $$
03

Calculate the x-component of the velocity of the center of mass

Since Body A is the only body moving in the x-direction, its velocity contributes to the x-component of the center of mass velocity. Use the formula for \(v_{cm}\), considering only the x-component: $$ v_{cm_x} = \frac{m_A v_{A_x}+ m_B v_{B_x}}{M} = \frac{3(14) + 4(0)}{7}=\frac{42}{7}=6 \ m/s $$
04

Calculate the y-component of the velocity of the center of mass

Since Body B is the only body moving in the y-direction, its velocity contributes to the y-component of the center of mass velocity. Use the formula for \(v_{cm}\), considering only the y-component: $$ v_{cm_y} = \frac{m_A v_{A_y}+ m_B v_{B_y}}{M} = \frac{3(0) + 4(-7)}{7}=\frac{-28}{7}=-4 \ m/s $$
05

Write the final answer

The x- and y-components of the velocity of the center of mass of the two bodies are: $$ v_{cm_x} = 6 \ m/s \\ v_{cm_y} = -4 \ m/s $$

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Most popular questions from this chapter

A \(58-\) kg astronaut is in space, far from any objects that would exert a significant gravitational force on him. He would like to move toward his spaceship, but his jet pack is not functioning. He throws a 720 -g socket wrench with a velocity of \(5.0 \mathrm{m} / \mathrm{s}\) in a direction away from the ship. After \(0.50 \mathrm{s}\), he throws a 800 -g spanner in the same direction with a speed of \(8.0 \mathrm{m} / \mathrm{s} .\) After another $9.90 \mathrm{s}\(, he throws a mallet with a speed of \)6.0 \mathrm{m} / \mathrm{s}$ in the same direction. The mallet has a mass of \(1200 \mathrm{g}\) How fast is the astronaut moving after he throws the mallet?
A projectile of mass \(2.0 \mathrm{kg}\) approaches a stationary target body at \(8.0 \mathrm{m} / \mathrm{s} .\) The projectile is deflected through an angle of \(90.0^{\circ}\) and its speed after the collision is $6.0 \mathrm{m} / \mathrm{s}$ What is the speed of the target body after the collision if the collision is perfectly elastic?
Consider two falling bodies. Their masses are \(3.0 \mathrm{kg}\) and $4.0 \mathrm{kg} .\( At time \)t=0,$ the two are released from rest. What is the velocity of their \(\mathrm{CM}\) at \(t=10.0 \mathrm{s} ?\) Ignore air resistance.
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