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 of the pin and (b) of the ball just after collision.

Verified step by step guidance

Identify the type of collision: This is a two-dimensional elastic collision, as the bowling ball and pin move in different directions after the collision. Both momentum and kinetic energy are conserved.

Set up the conservation of momentum equations: Momentum is conserved in both the x-direction (original direction of the ball) and the y-direction (perpendicular to the original direction). Let the mass of the pin be m and the mass of the ball be 5m. Let the final velocity of the pin be v_p and the final velocity of the ball be v_b. The equations are: mv=mv

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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 bowling scenario, the momentum of the bowling ball and the pin must be analyzed before and after the collision to determine their respective speeds.

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 is inelastic, meaning that while momentum is conserved, kinetic energy is not. This affects how we calculate the final velocities of the objects involved.

Vector Components

When analyzing collisions, it is essential to break down velocities into their vector components, typically along the x (horizontal) and y (vertical) axes. This allows for the application of conservation laws in each direction separately, facilitating the calculation of the final speeds and directions of the bowling ball and pin after the collision.

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Textbook Question

The gravitational slingshot effect. Figure 9–62 shows the planet Saturn moving in the negative 𝓍 direction at its orbital speed (with respect to the Sun) of 9.6 km/s. The mass of Saturn is 5.69 x 10²⁶ kg. A spacecraft with mass 825 kg approaches Saturn. When far from Saturn, it moves in the +𝓍 direction at 10.4 km/s. The gravitational attraction of Saturn (a conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. Using momentum conservation in one dimension, estimate the final speed of the spacecraft after it is far enough away to be considered free of Saturn’s gravitational pull. Assume the spacecraft does not affect the orbit of Saturn whose mass is so much larger.

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