Final answer:
The initial speed at which the ball was initially travelling was approximately 8.775 m/s, which can be calculated using the equations of uniformly accelerated motion and the given distance and time until the ball came to rest.
Step-by-step explanation:
To determine the initial speed at which the ball was initially travelling before it came to rest, we can use the equations of uniformly accelerated motion. In this case, the ball is decelerating (negative acceleration), which means we'll be considering the ball's deceleration until it stops.
Let d represent the distance the ball rolled before coming to rest, t be the time it took, a be the acceleration (which will have a negative value because it is a deceleration), u be the initial speed (what we're trying to find), and v be the final speed (which is 0 m/s since the ball comes to rest).
Since we know the final speed (v), the distance (d), and the time (t), we can use the equation:
\[v = u + at\]
Since v is 0 (the ball comes to rest), and we are looking for u, we rearrange to:
\[u = v - at\]
First, we need to calculate the acceleration (a) using the formula:
\[d = ut + \frac{1}{2}at^2\]
And since u = 0 at the end (v = 0), the formula simplifies to:
\[d = \frac{1}{2}at^2\]
So:
\[a = \frac{2d}{t^2}\]
Plugging the given values:
\[a = \frac{2 \times 32.9\,m}{(3.75\,s)^2}\]
By calculating, we find that:
a = -2.34\,m/s^2
This is the deceleration of the ball. We can now use this value of a to calculate the initial speed of the ball:
\[u = 0 - (-2.34\,m/s^2 \times 3.75\,s)\]
By calculating, we find that:
u = 8.775\,m/s
The initial speed at which the ball was initially travelling was approximately 8.775 m/s.