A billiard ball of mass m
 = 0.150 kg hits the cushion of a billiard table at an angle of θ1
 = 46.0° at a speed of v1
 = 2.40 m/s. It bounces off at an angle of θ2
 = 57.0° and a speed of v2
 = 2.20 m/s.

a) What is the magnitude of the change in momentum of the billiard ball?


b) In which direction does the change of momentum vector point? (Take the x-axis along the cushion and specify your answer in degrees.)

### Problem Breakdown:

We have a billiard ball of mass \( m = 0.150 \, \text{kg} \) hitting the cushion at a given angle and speed, and bouncing off at different angles and speeds. The goal is to find the **change in momentum** (both magnitude and direction).

To solve this, we will consider the momentum change in two perpendicular directions:
- The **x-direction** (parallel to the cushion).
- The **y-direction** (perpendicular to the cushion).

### Given Information:

- Mass of the ball: \( m = 0.150 \, \text{kg} \).
- Initial angle of incidence: \( \theta_1 = 46.0^\circ \).
- Initial speed: \( v_1 = 2.40 \, \text{m/s} \).
- Final angle after bouncing: \( \theta_2 = 57.0^\circ \).
- Final speed: \( v_2 = 2.20 \, \text{m/s} \).

### Approach:

The momentum of an object is given by:
\[
\vec{p} = m \vec{v}
\]

We'll decompose the velocities into components along the x-axis and y-axis, and find the change in momentum for each component.

### Part (a): Magnitude of the Change in Momentum

#### 1. Initial Momentum (\( \vec{p}_1 \)):

- **x-component** of initial velocity:
  \[
  v_{1x} = v_1 \cos \theta_1 = 2.40 \, \text{m/s} \cdot \cos 46.0^\circ
  \]
- **y-component** of initial velocity:
  \[
  v_{1y} = v_1 \sin \theta_1 = 2.40 \, \text{m/s} \cdot \sin 46.0^\circ
  \]

The corresponding initial momenta are:
\[
p_{1x} = m v_{1x}, \quad p_{1y} = m v_{1y}
\]

#### 2. Final Momentum (\( \vec{p}_2 \)):

- **x-component** of final velocity:
  \[
  v_{2x} = v_2 \cos \theta_2 = 2.20 \, \text{m/s} \cdot \cos 57.0^\circ
  \]
- **y-component** of final velocity:
  \[
  v_{2y} = v_2 \sin \theta_2 = 2.20 \, \text{m/s} \cdot \sin 57.0^\circ
  \]

The final momenta are:
\[
p_{2x} = m v_{2x}, \quad p_{2y} = m v_{2y}
\]

#### 3. Change in Momentum:

The change in momentum in each direction is:
\[
\Delta p_x = p_{2x} - p_{1x}, \quad \Delta p_y = p_{2y} - p_{1y}
\]

Finally, the magnitude of the total momentum change is:
\[
\Delta p = \sqrt{(\Delta p_x)^2 + (\Delta p_y)^2}
\]

#### Part (b): Direction of the Change in Momentum Vector

The direction of the change in momentum is given by:
\[
\theta_{\Delta p} = \tan^{-1} \left( \frac{\Delta p_y}{\Delta p_x} \right)
\]

Let me now calculate these values.### Solutions:

#### Part (a): Magnitude of the Change in Momentum
The magnitude of the change in momentum is:
\[
\Delta p = 0.0726 \, \text{kg} \cdot \text{m/s}
\]

#### Part (b): Direction of the Change in Momentum Vector
The direction of the change in momentum vector is:
\[
\theta_{\Delta p} = 165.8^\circ
\]
This is measured counterclockwise from the positive x-axis (along the cushion).

---

Would you like further details or clarification on any part of the process?

### Related Questions:
1. How does the mass of the billiard ball affect the momentum change?
2. What would happen to the momentum change if the bounce angles were smaller?
3. How does the speed of the ball influence the magnitude of the momentum change?
4. How does the direction of momentum relate to Newton’s third law in collisions?
5. Could friction between the ball and cushion affect these results?

### Tip:
Always break down momentum problems into x- and y-components, especially when dealing with angles, to simplify the calculations.

A billiard ball of mass m = 0.150 kg hits the cushion of a billiard table at an angle of θ1 = 46.0° at a speed of v1 = 2.40 m/s. It bounces off at an angle of θ2 = 57.0° and a speed of v2 = 2.20 m/s.

a) What is the magnitude of the change in momentum of the billiard ball?

b) In which direction does the change of momentum vector point? (Take the x-axis along the cushion and specify your answer in degrees.)

Learn how to calculate the change in momentum of a billiard ball hitting the cushion at an angle of 46.0° with a speed of 2.40 m/s and bouncing off at 57.0° with a speed of 2.20 m/s. This problem covers key physics concepts like momentum, vector decomposition, and trigonometry. Suitable for high school students.

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