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Chapter 9: Problem 32

A block of mass \(m\) undergoes a one-dimensional elastic collision with a block of mass \(M\) initially at rest. If both blocks have the same speed after colliding, how are their masses related?

Short Answer

Expert verified
The blocks must have equal masses \(m = M\), in order to have the same speed after the collision.

Step by step solution

01

Understand the conservation of momentum

By the law of conservation of momentum, the total initial momentum equals the total final momentum. If \(v\) is the initial speed of the block of mass \(m\) and \(u\) is the final speed of both blocks after collision then, initially, the momentum is \(mv\) and finally, the momentum is \(mu\) + \(Mu\). So, we can write this as \(mv = m u + M u\).
02

Solve for the final speed

From the equation \(m v = m u + M u\) in Step 1, we can factor out \(u\) from the RHS and then isolate \(u\). This yields \(u = \frac{m v}{m + M}\).
03

Understand Kinetic Energy Conservation

By the law of conservation of kinetic energy, the total initial kinetic energy equals the total final kinetic energy. The initial kinetic energy of mass \(m\) is \(\frac{1}{2} m v^2\) and the final kinetic energy of both blocks is \(\frac{1}{2} m u^2 + \frac{1}{2} M u^2\). So, we can write this as \(\frac{1}{2} m v^2 = \frac{1}{2} m u^2 + \frac{1}{2} M u^2\).
04

Insert the final speed into the kinetic energy equation

Substitute \(u = \frac{m v}{m + M}\) from Step 2 into the equation from Step 3 and simplify. We will get \(m v^2 = m \left(\frac{m v}{m + M}\right)^2 + M \left(\frac{m v}{m + M}\right)^2\). Simplification of this equation will lead to the relation between masses \(m\) and \(M\).
05

Solve for the mass relation

The equation from Step 4 simplifies to \(m v^2 = \frac{m^2 v^2}{m + M} + \frac{M m^2 v^2}{(m + M)^2}\). Upon further simplification, we find that all terms with \(v\) and \(v^2\) cancel out, indicating that the final velocity doesn't depend on the initial speed of block \(m\). Eventually we get \(m = M\).

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Most popular questions from this chapter

Give three everyday examples of inelastic collisions.

Consider a system of three equal-mass particles moving in a plane; their positions are given by \(a_{i} \hat{\imath}+b_{i} \hat{\jmath},\) where \(a_{i}\) and \(b_{i}\) are functions of time with the units of position. Particle 1 has \(a_{1}=3 t^{2}+5\) and \(b_{1}=0 ;\) particle 2 has \(a_{2}=7 t+2\) and \(b_{2}=2 ;\) particle 3 has \(a_{3}=3 t\) and \(b_{3}=2 t+6 .\) Find the center-of-mass position, velocity, and acceleration of the system as functions of time.

An alpha particle ( \(^{4} \mathrm{He}\) ) strikes a stationary gold nucleus \(\left(^{197} \mathrm{Au}\right)\) head-on. What fraction of the alpha's kinetic energy is transferred to the gold? Assume a totally elastic collision.

Mass \(m,\) moving at speed \(2 v,\) approaches mass \(4 m,\) moving at speed \(v .\) The two collide elastically head-on. Find expressions for their subsequent speeds.

Explosive bolts separate a \(950-\mathrm{kg}\) communications satellite from its \(640-\mathrm{kg}\) booster rocket, imparting a \(350-\mathrm{N} \cdot\) s impulse. At what relative speed do satellite and booster separate?
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