For one part of a chain-reaction machine that she is designing, a classmate will roll a ball with a mass of 1.2 kg down a ramp. At the bottom of the ramp, the ball will collide with a 1.8-kg pendulum block. Only the pendulum block will swing upward after the collision, and it must reach a maximum height of 0.075 m in order to continue the chain reaction. Determine the velocity of the ball just before it collides with the block in order for this part of the chain reaction machine to work.

To solve this problem, we’ll approach it by using the conservation of energy and conservation of momentum principles.

### Step 1: Calculate the Potential Energy of the Pendulum Block at Maximum Height
When the pendulum block swings upward to a maximum height \( h = 0.075 \, \text{m} \), it gains gravitational potential energy. We can calculate this energy using the equation:
\[
PE = m_{\text{block}} \cdot g \cdot h
\]
where:
- \( m_{\text{block}} = 1.8 \, \text{kg} \) (mass of the pendulum block),
- \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( h = 0.075 \, \text{m} \) (maximum height).

Substituting values:
\[
PE = 1.8 \cdot 9.81 \cdot 0.075 = 1.32315 \, \text{J}
\]

### Step 2: Determine the Required Velocity of the Pendulum Block Immediately After Collision
The kinetic energy of the pendulum block immediately after the collision must be equal to the potential energy calculated above, as this is the energy needed to reach the required height. Using the kinetic energy formula:
\[
KE = \frac{1}{2} m_{\text{block}} v_{\text{block}}^2
\]
where \( KE = PE = 1.32315 \, \text{J} \), and solving for \( v_{\text{block}} \):
\[
1.32315 = \frac{1}{2} \cdot 1.8 \cdot v_{\text{block}}^2
\]
\[
v_{\text{block}}^2 = \frac{1.32315 \cdot 2}{1.8} = 1.47
\]
\[
v_{\text{block}} = \sqrt{1.47} \approx 1.21 \, \text{m/s}
\]

So, the velocity of the pendulum block immediately after the collision must be approximately \( 1.21 \, \text{m/s} \).

### Step 3: Apply Conservation of Momentum to Find the Required Velocity of the Ball Before Collision
We assume a perfectly inelastic collision where the ball and block do not stick together but the block is set in motion with the required velocity. Using the conservation of momentum:
\[
m_{\text{ball}} v_{\text{ball}} = m_{\text{ball}} v_{\text{ball}}' + m_{\text{block}} v_{\text{block}}
\]


Learn how to calculate the velocity of a ball in a chain-reaction machine where it collides with a pendulum block. The pendulum block, after reaching a height of 0.075 m, must continue the reaction. This solution involves conservation of energy and momentum, making it suitable for students in Grades 11-12.

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