In order to convert a tough split in bowling, it is necessary to strike the pin a glancing blow as shown in Fig. 9–64. Assume that the bowling ball, traveling at 14.0 m/s just before it strikes the pin, has five times the mass of a pin and that the pin goes off at 75° from the original direction of the ball. Calculate the speed
(a) of the pin and (b) of the ball just after collision, and
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Verified step by step guidance

1. Identify the type of collision. Since the problem does not specify that kinetic energy is conserved, assume an elastic collision where both momentum and kinetic energy are conserved.
2. Set up the conservation of momentum equations in the x and y directions. Let the initial velocity of the pin be zero since it is stationary before being struck. Use the given angle (75 degrees) to resolve the velocities into components.
3. Apply the conservation of kinetic energy equation. Since it's an elastic collision, the total kinetic energy before and after the collision must be equal.
4. Solve the system of equations derived from the conservation laws to find the unknown velocities of the pin and the ball after the collision. You will have two equations from momentum conservation (one for each direction) and one from energy conservation.
5. Check the results for physical plausibility (e.g., speeds should be non-negative, and the direction of motion should make sense given the problem statement).

Key Concepts

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 bowling scenario, the momentum of the bowling ball and the pin must be analyzed before and after the collision to determine their respective speeds. This principle is crucial for solving problems involving collisions.

Elastic and Inelastic Collisions

Collisions can be classified as elastic or inelastic based on whether kinetic energy is conserved. In this case, the collision between the bowling ball and the pin can be treated as inelastic since the ball transfers some of its momentum to the pin, resulting in a change in their velocities. Understanding the nature of the collision helps in applying the correct equations to find the final speeds.

Vector Components

When analyzing collisions, it is essential to break down velocities into their vector components, typically along the x and y axes. In this problem, the pin moves at an angle of 75° after the collision, necessitating the use of trigonometric functions to resolve its velocity into components. This approach allows for the application of conservation laws in multiple dimensions.