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Ch 11: Impulse and Momentum
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 11: Impulse and MomentumProblem 49a
Chapter 11, Problem 49a

A bullet of mass m is fired into a block of mass M that is at rest. The block, with the bullet embedded, slides distance d across a horizontal surface. The coefficient of kinetic friction is μk .Find an expression for the bullet's speed vbullet.

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Start by applying the principle of conservation of momentum for the collision. Since the bullet embeds itself into the block, this is an inelastic collision. The total momentum before the collision is the momentum of the bullet, which is \( p_{initial} = m \cdot v_{bullet} \). After the collision, the combined mass \( (m + M) \) moves with a velocity \( v_{final} \). Thus, the momentum conservation equation is: \( m \cdot v_{bullet} = (m + M) \cdot v_{final} \).
Next, determine the velocity \( v_{final} \) of the block and bullet system immediately after the collision. Rearrange the momentum conservation equation to solve for \( v_{final} \): \( v_{final} = \frac{m \cdot v_{bullet}}{m + M} \).
Now, analyze the motion of the block and bullet system as it slides across the surface. The work done by the kinetic friction force \( f_k \) is equal to the loss of kinetic energy of the system. The frictional force is given by \( f_k = \mu_k \cdot (m + M) \cdot g \), where \( g \) is the acceleration due to gravity. The work-energy principle states: \( \frac{1}{2} (m + M) \cdot v_{final}^2 = f_k \cdot d \).
Substitute the expressions for \( f_k \) and \( v_{final} \) into the work-energy equation. This gives: \( \frac{1}{2} (m + M) \cdot \left( \frac{m \cdot v_{bullet}}{m + M} \right)^2 = \mu_k \cdot (m + M) \cdot g \cdot d \).
Simplify the equation to isolate \( v_{bullet} \). Cancel out \( (m + M) \) where appropriate, and solve for \( v_{bullet} \): \( v_{bullet} = \sqrt{\frac{2 \cdot \mu_k \cdot g \cdot d \cdot (m + M)}{m}} \). This is the final expression for the bullet's speed.

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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, provided no external forces act on it. In this scenario, when the bullet embeds into the block, we can apply this principle to relate the bullet's initial speed to the combined speed of the block and bullet after the collision.
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Frictional Force

Frictional force is the resistance that one surface or object encounters when moving over another. It is calculated as the product of the normal force and the coefficient of kinetic friction (F_friction = μ_k * N). In this problem, the frictional force acts on the block and bullet system as they slide across the surface, opposing their motion and affecting their deceleration.
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Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by friction as the block slides a distance d will equal the initial kinetic energy of the bullet-block system after the collision. This relationship allows us to derive the bullet's initial speed based on the distance slid and the frictional force.
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Related Practice
Textbook Question

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