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Chapter 7: Problem 42

A \(0.020-\mathrm{kg}\) bullet traveling at \(200.0 \mathrm{m} / \mathrm{s}\) east hits a motionless \(2.0-\mathrm{kg}\) block and bounces off it, retracing its original path with a velocity of \(100.0 \mathrm{m} / \mathrm{s}\) west. What is the final velocity of the block? Assume the block rests on a perfectly frictionless horizontal surface.

Short Answer

Expert verified
Solution: Use the law of conservation of linear momentum to determine the final velocity of the block (v鈧俧) after the collision. Calculate the initial and final momentums of the bullet and the initial momentum of the block. Apply the conservation of linear momentum law and solve for the final momentum of the block (p鈧俧). Finally, calculate the final velocity of the block (v鈧俧) using the equation p鈧俧 = m鈧倂鈧俧.

Step by step solution

01

Understand the conservation of linear momentum law

The law of conservation of linear momentum states that the total linear momentum of a closed system remains constant, provided no external forces act on it. In our problem, the forces acting are in the horizontal direction, so we will focus on the conservation of momentum along the x-axis.
02

Calculate the initial momentum of bullet

Given the bullet's mass \(m_1 = 0.020 \, kg\) and its initial velocity \(v_{1i} = 200.0 \, m/s\) to the east, we can calculate the bullet's initial momentum to be: \(p_{1i} = m_1 v_{1i}\)
03

Calculate the initial momentum of the block

Since the block is initially at rest, its initial momentum is \(0\). Let \(m_2 = 2.0 \, kg\) be the mass of the block, and \(p_{2i} = m_2 v_{2i} = 0\) be its initial momentum.
04

Calculate the final momentum of the bullet

The bullet's final velocity is given as \(v_{1f} = -100.0 \, m/s\) to the west (because it retraces its original path). We can calculate the bullet's final momentum to be \(p_{1f} = m_1 v_{1f}\).
05

Apply the conservation of linear momentum law

To determine the final velocity of the block \((v_{2f})\) , we apply the conservation of linear momentum law, i.e., the total initial momentum equals the total final momentum: \(p_{1i} + p_{2i} = p_{1f} + p_{2f}\). Rearranging the equation to solve for \(p_{2f}\): \(p_{2f} = (p_{1i} + p_{2i}) - p_{1f}\). Substitute the known values for \(p_{1i}\), \(p_{2i}\), and \(p_{1f}\) to find \(p_{2f}\).
06

Calculate the final velocity of the block

Since we have the final momentum of the block \((p_{2f})\), we can find the final velocity of the block \((v_{2f})\) by using the equation: \(p_{2f} = m_2 v_{2f}\). Solve for \(v_{2f}\) by dividing the equation by the block's mass \(m_2\). Following these steps will give you the final velocity of the block after the collision.

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