To determine the acceleration of the ball, we can use the kinematic equation which relates distance, initial velocity, time, and acceleration:
[tex]\[ d = v_i t + \frac{1}{2} a t^2 \][/tex]
Here:
- [tex]\( d \)[/tex] is the distance traveled, which is 25 meters.
- [tex]\( v_i \)[/tex] is the initial velocity, which is 0 m/s since the ball starts from rest.
- [tex]\( t \)[/tex] is the time taken, which is 4.6 seconds.
- [tex]\( a \)[/tex] is the acceleration that we need to find.
Given the initial velocity [tex]\( v_i = 0 \)[/tex] m/s, the equation simplifies to:
[tex]\[ d = \frac{1}{2} a t^2 \][/tex]
We can rearrange the equation to solve for acceleration [tex]\( a \)[/tex]:
[tex]\[ a = \frac{2d}{t^2} \][/tex]
Now we can plug in the given values:
- [tex]\( d = 25 \)[/tex] meters
- [tex]\( t = 4.6 \)[/tex] seconds
So,
[tex]\[ a = \frac{2 \times 25}{(4.6)^2} \][/tex]
Calculate the term in the denominator first:
[tex]\[ (4.6)^2 = 21.16 \][/tex]
Then,
[tex]\[ a = \frac{50}{21.16} \][/tex]
Upon calculating,
[tex]\[ a \approx 2.362948960302458 \][/tex]
Therefore, the acceleration of the ball is approximately:
[tex]\[ a \approx 2.36 \, \text{m/s}^2 \][/tex]
