Week9

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University of California, Berkeley *

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Mechanical Engineering

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Oct 30, 2023

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Uploaded by DrCamelMaster579

Week 9 Problem 1-Two sliding blocks are set in motion on a horizontal frictionless track. A spring of force constant k is attached to the front of a first block of mass m1, which moves to right with a speed v1. The second block with a mass m2 moves at a slower speed v2 to the right, as shown in the figure. The spring will compress against m2. a) When the spring is at its maximum compression, what is v1 − v2? Your answer should be simple. b) Find the speed of each mass at maximum compression. c) Find the maximum compression xmax of the spring. d) Explain whether the collision is elastic or not. e) If k were very large, what would this collision be like? Is it still elastic/inelastic, and how would the motion during the collision change? This is a conceptual question, but use quantitative proof for your conclusion if possible.

Problem 2-Consider three spheres lined up in a row along the x-axis with masses m1 = M, m2 = M/2, and m3 = M. The sphere with m = m1 has an initial velocity V0 in the +x direction, while the other two spheres are initially at rest. The sphere with m = m1 has an elastic collision with m2 and then m2 collides and sticks to m3. a) Determine the velocities of m1 and m2 after the first collision. [10 pts.] b) Determine the velocities of m2 and m3 after the second collision. [5 pts.] c) After the second collision will m1 collide with the m2, m3 combination? If they do not collide, for what values of m3 would this collision occur? [10 pts.]

Problem 3- Consider a block launched from a wedge of height H and angle θ by a spring with spring constant k. The spring is compressed from its equilibrium by a distance d. A block with mass m is placed on the compressed spring halfway on the incline of the wedge with a height H/2 a) What is the velocity of the block as it leaves the hill? Include the effect of friction with coefficient of kinetic friction µ. Assume the spring is strong enough to launch the block from the wedge. [10 pts.] b) What is the velocity of the block when it falls back to a height H/2? [5 pts.] c) If at that point in part (b) the block breaks into two pieces with mass 1/3 m and 2/3 m such that the 1/3 m piece falls straight down with the same vertical velocity as before, how far apart will the two pieces be when they land? For the answer to this section, you do not have to plug in your values from parts (a) and (b). [10 pts.]

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Problem 4-(25 points) Suppose a spacecraft with initial mass of mi. Without its propellant, the spacecraft has a mass of mf = mi/3. The rocket that powers the spacecraft is designed to eject the propellant with a speed of u relative to the rocket at a constant rate of R. The spacecraft is initially at rest in space and travels in a straight line. a) How long would it take the rocket to release all its propellant? b) What is m(t), the mass of the rocket as a function of time? c) What is v(t), the speed of the rocket as a function of time? d) How far will the spacecraft travel before its rocket uses all the propellant and shuts down? Hint: ∫ ln(1 − ࠵?࠵?) ࠵?࠵? = ࠵?࠵?−1 ࠵? ln(1 − ࠵?࠵?) − ࠵?