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Chapter 8: Problem 69

Two masses, \(m_{1}=2.0 \mathrm{~kg}\) and \(m_{2}=3.0 \mathrm{~kg}\), are moving in the \(x y\) -plane. The velocity of their center of mass and the velocity of mass 1 relative to mass 2 are given by the vectors \(v_{\mathrm{cm}}=(-1.0,+2.4) \mathrm{m} / \mathrm{s}\) and \(v_{\mathrm{rel}}=(+5.0,+1.0) \mathrm{m} / \mathrm{s} .\) Determine a) the total momentum of the system b) the momentum of mass 1 , and c) the momentum of mass 2 .

Short Answer

Expert verified
The total momentum of the system is (-5.0, +12.0) kg.m/s. b) What is the momentum of mass 1? The momentum of mass 1 is (+6.0, -4.0) kg.m/s. c) What is the momentum of mass 2? The momentum of mass 2 is (-11.0, +16.0) kg.m/s.

Step by step solution

01

a) Total momentum of the system

To find the total momentum of the system (P_total), we will first add the masses together (M) and multiply it by the given center of mass velocity (v_cm): \(P_{\text{total}} = M \times v_{\text{cm}}\) \(M = m_1 + m_2 = 2.0 kg + 3.0 kg = 5.0 kg\) \(v_{\text{cm}} = (-1.0,+2.4) \frac{m}{s}\) So, \(P_{\text{total}} = (5.0 kg) \times (-1.0,+2.4) \frac{m}{s} = (-5.0,+12.0) \frac{kg.m}{s}\)
02

b) Momentum of mass 1

To find the momentum of mass 1 (P_1), we need to first find its velocity (v_1). To calculate this, we will use the relative velocity of mass 1 to mass 2 (v_rel) and the center of mass velocity (v_cm). We know that: \(v_{\text{rel}} = v_1 - v_2\) \(v_1 = v_{\text{rel}} + v_2\) Now we can find the velocity of mass 2 (v_2) using the center of mass velocity: \(v_{\text{cm}} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}\) \(v_2 = \frac{v_{\text{cm}}(m_1 + m_2) - m_1 v_1}{m_2}\) Substitute the values: \(v_2 = \frac{(-1.0,+2.4)(5.0) - 2.0 v_1}{3.0}\) \(v_2 = \frac{(-5.0,+12.0) - 2.0 v_1}{3.0}\) Now, substitute \(v_2\) back into the equation for \(v_1\): \(v_1 = (+5.0,+1.0) + \frac{(-5.0,+12.0) - 2.0 v_1}{3.0}\) Solving for \(v_1\), we get: \(v_1 = (+3.0,-2.0) \frac{m}{s}\) Now, we can find the momentum of mass 1 by multiplying its mass with its velocity: \(P_1 = m_1 \times v_1 = (2.0 kg) \times (+3.0,-2.0) \frac{m}{s} = (+6.0,-4.0) \frac{kg.m}{s}\)
03

c) Momentum of mass 2

Now that we have the momentum of mass 1, we can use the total momentum of the system to determine the momentum of mass 2 (P_2): \(P_{\text{total}} = P_1 + P_2\) \(P_2 = P_{\text{total}} - P_1 = (-5.0,+12.0) \frac{kg.m}{s} - (+6.0,-4.0) \frac{kg.m}{s} = (-11.0,+16.0) \frac{kg.m}{s}\)

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Most popular questions from this chapter

Suppose you place an old-fashioned hourglass, with sand in the bottom, on a very sensitive analytical balance to determine its mass. You then turn it over (handling it with very clean gloves) and place it back on the balance. You want to predict whether the reading on the balance will be less than, greater than, or the same as before. What do you need to calculate to answer this question? Explain carefully what should be calculated and what the results would imply. You do not need to attempt the calculation.

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You are piloting a spacecraft whose total mass is \(1000 \mathrm{~kg}\) and attempting to dock with a space station in deep space. Assume for simplicity that the station is stationary, that your spacecraft is moving at \(1.0 \mathrm{~m} / \mathrm{s}\) toward the station, and that both are perfectly aligned for docking. Your spacecraft has a small retro-rocket at its front end to slow its approach, which can burn fuel at a rate of \(1.0 \mathrm{~kg} / \mathrm{s}\) and with an exhaust velocity of \(100 \mathrm{~m} / \mathrm{s}\) relative to the rocket. Assume that your spacecraft has only \(20 \mathrm{~kg}\) of fuel left and sufficient distance for docking. a) What is the initial thrust exerted on your spacecraft by the retro-rocket? What is the thrust's direction? b) For safety in docking, NASA allows a maximum docking speed of \(0.02 \mathrm{~m} / \mathrm{s}\). Assuming you fire the retro-rocket from time \(t=0\) in one sustained burst, how much fuel (in kilograms) has to be burned to slow your spacecraft to this speed relative to the space station? c) How long should you sustain the firing of the retrorocket? d) If the space station's mass is \(500,000 \mathrm{~kg}\) (close to the value for the ISS), what is the final velocity of the station after the docking of your spacecraft, which arrives with a speed of \(0.02 \mathrm{~m} / \mathrm{s}\) ?
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