A sphere is fired downwards into a medium with an initial speed of 40 .

Part A
If it experiences a deceleration of , where is in seconds, determine the distance traveled before it stops.

To determine the distance traveled by the sphere before it stops, we need to use the equation of motion:

\(v^2 = u^2 + 2as\)

Where:
- \(v\) is the final velocity (which is 0 in this case because it stops)
- \(u\) is the initial velocity (40 m/s)
- \(a\) is the deceleration (\(-k\) as given in the problem)
- \(s\) is the distance traveled

Plugging in the given values, we have:

\(0 = (40)^2 + 2(-k)s\)

Simplifying:

\(0 = 1600 - 2ks\)

Rearranging the equation to solve for \(s\):

\(2ks = 1600\)

\(s = \frac{1600}{2k}\)

So, the distance traveled by the sphere before it stops is \( \frac{1600}{2k} \), where \(k\) is the given deceleration in seconds.