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: $m_{\text{ball}} v_{\text{ball}} = m_{\text{ball}} v_{\text{ball}}' + m_{\text{block}} v_{\text{block}}$