To find the average acceleration of the ball while it is in contact with the wall, we can use the formula for acceleration:
\[ a = \frac{\Delta v}{\Delta t} \]
where \(\Delta v\) is the change in velocity and \(\Delta t\) is the time duration of contact with the wall.
Step 1: Determine the initial and final velocities.
- The initial velocity \(v_i\) of the ball is \(20 , \text{m/s}\) (toward the wall, which we will consider as positive).
- The final velocity \(v_f\) after the ball rebounds is \(-14.76 , \text{m/s}\) (away from the wall, which we consider as negative).
Step 2: Calculate the change in velocity \(\Delta v\).
\[ \Delta v = v_f - v_i = -14.76 , \text{m/s} - 20 , \text{m/s} = -14.76 , \text{m/s} - 20 , \text{m/s} = -34.76 , \text{m/s} \]
Step 3: Use the time duration of contact.
Given: \(\Delta t = 0.00903 , \text{s}\)
Step 4: Calculate the average acceleration \(a\).
\[ a = \frac{\Delta v}{\Delta t} = \frac{-34.76 , \text{m/s}}{0.00903 , \text{s}} \approx -3846.76 , \text{m/s}^2 \]
Step 5: Final answer and interpretation.
The average acceleration of the ball while it is in contact with the wall is approximately:
\[ \boxed{-3846.76} , \text{m/s}^2 \]
(Note: The negative sign indicates that the acceleration is in the direction opposite to the initial motion, which makes sense as it is decelerating upon striking the wall and then reversing direction.)