A 22-g bullet traveling 240 m/s penetrates a 2.0-kg block of wood and emerges going 130 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges?

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Step 1: Identify the principle of conservation of momentum, which states that the total momentum before and after the collision is conserved. This applies because there are no external forces acting on the system.

Step 2: Write the equation for the total momentum before the collision. The bullet's initial momentum is given by $ p_{bullet, initial} = m_{bullet} \cdot v_{bullet, initial} $, where $ m_{bullet} = 0.022 \, \text{kg} $ and $ v_{bullet, initial} = 240 \, \text{m/s} $. The block is stationary, so its initial momentum is zero.

Step 3: Write the equation for the total momentum after the collision. The bullet's final momentum is $ p_{bullet, final} = m_{bullet} \cdot v_{bullet, final} $, where $ v_{bullet, final} = 130 \, \text{m/s} $. The block's final momentum is $ p_{block, final} = m_{block} \cdot v_{block, final} $, where $ m_{block} = 2.0 \, \text{kg} $ and $ v_{block, final} $ is the unknown velocity we need to solve for.

Step 4: Set up the conservation of momentum equation: $ p_{bullet, initial} + p_{block, initial} = p_{bullet, final} + p_{block, final} $. Substitute the known values into the equation: $ (0.022 \cdot 240) + 0 = (0.022 \cdot 130) + (2.0 \cdot v_{block, final}) $.

Step 5: Solve for $ v_{block, final} $ algebraically. Rearrange the equation to isolate $ v_{block, final} $: $ v_{block, final} = \frac{(0.022 \cdot 240) - (0.022 \cdot 130)}{2.0} $. Perform the arithmetic operations to find the block's final velocity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event. In this scenario, the bullet and the block form a system where the initial momentum (bullet's momentum) must equal the final momentum (combined momentum of the bullet and block) after the bullet passes through the block.

Momentum Calculation

Momentum is calculated as the product of an object's mass and its velocity (p = mv). For the bullet and the block, we will calculate their individual momenta before and after the bullet passes through the block. This calculation is essential to determine how the block's velocity changes as a result of the bullet's interaction.

Kinematics of Collisions

Kinematics of collisions involves analyzing the motion of objects before and after they collide or interact. In this case, we need to consider the initial velocities of both the bullet and the block, as well as the final velocity of the bullet after it exits the block, to find the resulting velocity of the block using the conservation of momentum.

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