We start with the equation of kinematic motion: y(t) = v0*t+ g*t^2. Here, y is the height of the ball from the bottom of the lake, t is time, v0 is the initial velocity, and g is the gravitational acceleration constant.
(1) The ball is dropped from a height of y=6.31+x, where x represents the depth of the lake. The ball is initially dropped, so the initial velocity v0 = 0, and travels the distance y in 5.05s.
So, we know that (6.31+x) = g*(5.05)2. By plugging in the magnitude of the gravitational acceleration g, you can solve for the depth of the lake, x.
(2) It helps to remember that acceleration is defined as the change in velocity over time. Thus, the velocity over time can be written as v(t) = v0 + a*t. Again, the initial velocity is zero, and the final velocity is vf = g*(5.05).
Because the velocity increases linearly over time, the average velocity over that time is just half the difference between the initial and final velocities: (vf-v0)/2. SInce the initial velocity is zero, the mean velocity is just half of the final velocity.
(3) Careful, this is a trick question! In free fall, the ball already falls from the diving board to the lake’s bottom in 5.05 seconds. Any nonzero initial velocity in the vertical direction would change that amount of time - this general fact can be confirmed with the equations above. Therefore, the magnitude of the initial velocity is still zero.
Of course, if the person throws the ball exactly in the horizontal direction, then they can throw it as fast or as slow as they like, without changing the time it takes to reach the bottom of the lake!
Hope this helps,
Sam
