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A block of mass M is moving at speed v0 on a frictionless surface that ends in a rigid
Chapter 9, Problem 93(choose chapter or problem)
A block of mass M is moving at speed v0 on a frictionless surface that ends in a rigid wall, heading toward a stationary block of mass nM, where \(n \geq 1\) (Fig. 9.30). Collisions between the two blocks or the left-hand block and the wall are elastic and one-dimensional.
(a) Show that the blocks will undergo only one collision with each other if \(n \leq 3\).
(b) Show that the blocks will undergo two collisions with each other if n = 4. (c) How many
collisions will the blocks undergo if n = 10, and what will be their final speeds?
Questions & Answers
QUESTION:
A block of mass M is moving at speed v0 on a frictionless surface that ends in a rigid wall, heading toward a stationary block of mass nM, where \(n \geq 1\) (Fig. 9.30). Collisions between the two blocks or the left-hand block and the wall are elastic and one-dimensional.
(a) Show that the blocks will undergo only one collision with each other if \(n \leq 3\).
(b) Show that the blocks will undergo two collisions with each other if n = 4. (c) How many
collisions will the blocks undergo if n = 10, and what will be their final speeds?
Step 1 of 9
(a) The second block is initially at rest. Therefore, the velocity of the first block after the collision is expressed as,
\(v_{1 \mathrm{f}}=\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right) v_{1 \mathrm{i}}\)
For \(m_{1}=M\), \(m_{2}=n M\) and \(v_{1 \mathrm{i}}=v_{0}\).
\(\begin{aligned} v_{1 f} & =\left(\frac{M-n M}{M+n M}\right) v_{0} \\ & =\frac{M(1-n)}{M(1+n)} v_{0} \\ & =\frac{1-n}{1+n} v_{0} \end{aligned}\)