Bob has just finished climbing a sheer cliff above a beach, and wants to figure out how high he climbed. All he has to use, however, is a baseball, a stopwatch, and a friend on the ground below with a long measuring tape. Bob is a pitcher, and knows that the fastest he can throw the ball is about 32.5 m/s. Bob starts the stopwatch as he throws the ball (with no way to measure the ball\'s initial trajectory), and watches carefully. The ball rises and then falls, and after 0.910 seconds the ball is once again level with Bob. Bob can\'t see well enough to time when the ball hits the ground. Bob\'s friend then measures that the ball landed 123 m from the base of the cliff. How high up is Bob, if the ball started from exactly 2 m above the edge of the cliff?

To determine how high Bob climbed, we can use the equation of motion for vertical projectile motion:

h = -0.5 * g * t^2 + v0y * t + y0

Where:
h is the height above the ground
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time it takes for the ball to reach the same level as Bob
v0y is the vertical component of the initial velocity of the ball
y0 is the initial height of the ball above the ground

First, let's calculate the time it takes for the ball to reach the same level as Bob. We know that it takes 0.910 seconds for the ball to do so.

Next, let's calculate the vertical component of the initial velocity of the ball, v0y. Since the ball was thrown horizontally, the vertical component of the initial velocity is equal to zero. This is because there is no initial upward or downward velocity.

Now, we know that Bob threw the ball horizontally with a speed of 32.5 m/s. Horizontal motion does not affect the ball's vertical position, so we don't need to consider the initial velocity for the vertical direction.

Finally, let's calculate the initial height of the ball, y0. We know that the ball started from exactly 2 m above the edge of the cliff.

Now, let's plug in the values into the equation:

h = -0.5 * 9.8 * (0.910)^2 + 0 * (0.910) + 2

Simplifying the equation:

h = -0.5 * 9.8 * 0.8281 + 0 + 2
h = -4.0409 + 2
h ≈ -2.0409

The negative value indicates that Bob's height above the ground is below the initial height of the ball. To find the absolute height above the ground, we multiply it by -1:

h ≈ -(-2.0409)
h ≈ 2.0409

Bob climbed approximately 2.0409 meters above the edge of the cliff.