A ball is moving perpendicularly to a wall approaching it with an initial speed of 22.8 m/s. After striking the wall it bounces in the opposite direction with a speed of 21.2 m/s. Taking the positive direction towards the wall, find the initial velocity of the ball. Note: pay attention to the sign of the velocity and acceleration.

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!

Find the initial velocity of a ball moving towards a wall with a speed of 22.8 m/s and bouncing back with a speed of 21.2 m/s. This kinematic problem involves velocity calculations and directional sign conventions, suitable for students in Grades 10-12.

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