Answer:
a) To find the depth of the lake, we can use the following equation:
distance = initial velocity x time + (1/2) x acceleration x time^2
As the ball sinks to the bottom with the same constant velocity, we can assume that the acceleration is due to gravity, which is 9.8 m/s^2 (going downward)
We know that the time it takes for the ball to hit the bottom is 5.05 seconds, so we can substitute these values into the equation:
distance = 0 x 5.05 s + (1/2) x 9.8 m/s^2 x (5.05 s)^2
distance = 12.6125 m
So the lake is 12.6125 m deep
b) To find the magnitude of the average velocity of the ball for the entire fall, we can use the equation:
average velocity = distance / time
We know that the distance is the depth of the lake, which is 12.6125 m, and the time is 5.05 s, so we can substitute these values into the equation:
average velocity = 12.6125 m / 5.05 s
average velocity = 2.49 m/s
c) To find the magnitude of the initial velocity of the ball when it is thrown, we can use the following equation:
initial velocity = final velocity^2 + 2 x acceleration x distance
As the ball reaches the bottom in the same time as when it was dropped, 5.05 s, we can assume that the final velocity is 0 m/s. We know that the acceleration is 9.8 m/s^2 (going downward), and the distance is the depth of the lake, which is 12.6125 m. So we can substitute these values into the equation:
initial velocity = 0^2 + 2 x 9.8 m/s^2 x 12.6125 m
initial velocity = 24.6 m/s
Note that the magnitude of initial velocity is 24.6 m/s, which is a scalar value, it doesn't have any direction.