A billiard ball of mass m
 = 0.150 kg hits the cushion of a billiard table at an angle of θ1
 = 50.0° at a speed of v1
 = 2.60 m/s. It bounces off at an angle of θ2
 = 59.0° and a speed of v2
 = 2.10 m/s.

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

To find the magnitude of the change in momentum, we need to calculate the momentum before and after the collision, and then find the vector difference between them.

### Momentum Before and After Collision

The momentum \( \vec{p} \) is given by:
\[
\vec{p} = m \vec{v}
\]
where \( m \) is the mass of the object and \( \vec{v} \) is the velocity vector. Since we have angles, we will decompose the velocities into their \( x \)- and \( y \)-components.

Given:
- Mass, \( m = 0.150 \, \text{kg} \)
- Initial speed, \( v_1 = 2.60 \, \text{m/s} \)
- Initial angle, \( \theta_1 = 50.0^\circ \)
- Final speed, \( v_2 = 2.10 \, \text{m/s} \)
- Final angle, \( \theta_2 = 59.0^\circ \)

#### 1. Calculate Initial Momentum Components
The initial momentum components (\( p_{1x} \) and \( p_{1y} \)) are:
\[
p_{1x} = m v_1 \cos(\theta_1)
\]
\[
p_{1y} = m v_1 \sin(\theta_1)
\]
Substitute the known values:
\[
p_{1x} = 0.150 \times 2.60 \times \cos(50.0^\circ)
\]
\[
p_{1y} = 0.150 \times 2.60 \times \sin(50.0^\circ)
\]

#### 2. Calculate Final Momentum Components
Similarly, for the final momentum components (\( p_{2x} \) and \( p_{2y} \)):
\[
p_{2x} = m v_2 \cos(\theta_2)
\]
\[
p_{2y} = m v_2 \sin(\theta_2)
\]
Substitute the values:
\[
p_{2x} = 0.150 \times 2.10 \times \cos(59.0^\circ)
\]
\[
p_{2y} = 0.150 \times 2.10 \times \sin(59.0^\circ)
\]

#### 3. Change in Momentum Components
The change in momentum components are:
\[
\Delta p_x = p_{2x} - p_{1x}
\]
\[
\Delta p_y = p_{2y} - p_{1y}
\]

#### 4. Magnitude of the Change in Momentum
The magnitude of the change in momentum is:
\[
|\Delta \vec{p}| = \sqrt{(\Delta p_x)^2 + (\Delta p_y)^2}
\]

I'll calculate these values next.The magnitude of the change in momentum of the billiard ball is approximately \( 0.093 \, \text{kg} \cdot \text{m/s} \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the result change if the angles were different?
2. What happens to the momentum if the ball's mass increases?
3. How do you calculate the direction of the change in momentum?
4. How is momentum conserved in such a collision?
5. What is the significance of the ball's final speed being lower than the initial speed?

**Tip:** When dealing with momentum changes in two dimensions, always break the momentum into components to make vector calculations easier.

A billiard ball of mass m = 0.150 kg hits the cushion of a billiard table at an angle of θ1 = 50.0° at a speed of v1 = 2.60 m/s. It bounces off at an angle of θ2 = 59.0° and a speed of v2 = 2.10 m/s. What is the magnitude of the change in momentum of the billiard ball?

Explore how to calculate the magnitude of the change in momentum when a billiard ball hits the cushion and bounces off at different angles and speeds. The problem covers vector decomposition, momentum formulas, and the application of momentum conservation in two dimensions. Suitable for high school physics students (Grades 11-12).

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