Given:

• The ball's initial speed towards the wall: $22.8 \, \text{m/s}$
• The ball bounces back with a speed of $21.2 \, \text{m/s}$
• The positive direction is towards the wall.

Step-by-Step Solution

1. Since the positive direction is towards the wall:

   • The initial velocity ($v_i$) is towards the wall. Therefore, it is positive: $v_i = +22.8 \, \text{m/s}$

2. When the ball bounces back, it moves in the opposite direction to the wall. Thus, its velocity is negative: $v_f = -21.2 \, \text{m/s}$

Answer

The initial velocity of the ball, considering the direction convention given, is: $v_i = +22.8 \, \text{m/s}$

Would you like more details or have any questions about this solution?

Here are 5 related questions for further exploration:

1. How is the change in velocity calculated for the ball after it strikes the wall?
2. What is the average acceleration of the ball during its contact with the wall?
3. How would the problem change if the wall were moving towards or away from the ball?
4. What would be the impulse imparted to the ball by the wall?
5. How does the coefficient of restitution relate to the velocities before and after the collision?

Tip: Always keep track of directions when working with vectors, as signs indicate direction in physics problems!