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 91.0 mph. 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 453 ft from the base of the cliff. How high up is Bob, if the ball started from exactly 5 ft above the edge of the cliff?
I assume that "level with Bob" means at the same height as it was thrown.
If he threw at a speed of 91.0 mi/hr = 133.467 ft/s, at an angle θ, then
the horizontal speed was 133.467 cosθ
Now, plugging in our distance traveled to landing,
133.467 cosθ t = 453
so, it took 453/(133.467 cosθ) seconds to land.
That means that if the cliff's height is h,
y(t) = h+5 + 133.467sinθ t - 16t^2
Now, at time t=0.910, y is again h+5, so
133.467 sinθ * 0.910 - 16(0.910)^2 = 0
sinθ = (16*.910)/133.467
θ = 6.263°
Thus,
y(t) = h+5 + 14.56t - 16t^2
The horizontal speed was 133.467 cos6.263° = 132.670 ft/s
So, it took 453/132.670 = 3.414 seconds to land.
So, now we know that
h+5 + 14.56*3.414 - 16*3.414^2 = 0
h = 131.778 ft
To determine how high Bob climbed, we can use the equations of motion for projectile motion. We'll need to find the initial vertical velocity of the baseball, which can be determined from the time it takes for the ball to reach its peak.
1. Convert the speed of the baseball from mph to ft/s:
91.0 mph * (5280 ft/mile) * (1/3600 h/s) = 134.13 ft/s
2. Calculate the time it takes for the baseball to reach its peak:
The time to reach the peak can be determined by dividing the total time by 2, as the ball reaches its maximum height halfway through its total flight time.
t = 0.910 s / 2 = 0.455 s
3. Calculate the initial vertical velocity of the baseball:
The initial vertical velocity (v_y0) can be calculated using the formula: v_y0 = g * t, where g is the acceleration due to gravity (32.2 ft/s²).
v_y0 = 32.2 ft/s² * 0.455 s = 14.66 ft/s
4. Calculate the maximum height reached by the baseball:
The maximum height (h_max) can be calculated using the formula: h_max = (v_y0)² / (2 * g).
h_max = (14.66 ft/s)² / (2 * 32.2 ft/s²) = 6.66 ft
5. Calculate the total time of flight for the baseball:
The total time (t_total) can be calculated as twice the time it took to reach the peak: t_total = 2 * 0.455 s = 0.91 s
6. Calculate the time it takes for the baseball to hit the ground:
Since the total time of flight is 0.91 seconds, and Bob observes the ball being level with him at 0.910 seconds, it means the ball took an additional time of (0.91 s - 0.910 s) = 0 seconds to reach the ground.
7. Calculate the horizontal distance traveled by the baseball:
The horizontal distance traveled (d) can be calculated using the formula: d = v_x * t, where v_x is the horizontal velocity of the ball.
Since there is no acceleration horizontally, the horizontal velocity remains constant:
d = 91.0 mph * (5280 ft/mile) * (1/3600 h/s) * 0.910 s = 116.82 ft
8. Calculate Bob's height above the cliff:
Bob's height above the cliff is the sum of the maximum height reached by the baseball (h_max) and the initial height above the edge of the cliff (5 ft):
Total height = h_max + 5 ft = 6.66 ft + 5 ft = 11.66 ft
Therefore, Bob climbed a total height of approximately 11.66 ft.
To determine how high up Bob is, we can break down the problem into two parts:
1. Calculate the time it takes for the ball to hit the ground.
2. Use the time calculated in step 1 to find the vertical distance the ball traveled and, thus, the height of the cliff.
Let's start with step 1:
Since Bob cannot see when the ball hits the ground, he needs to use the stopwatch to measure the time it takes for the ball to reach the level position once again. In this case, the ball is level with Bob after 0.910 seconds of throwing.
Now, let's move on to step 2:
To find the vertical distance the ball traveled, we need to calculate the time it takes to reach the maximum height. Since the ball is thrown vertically upwards, it follows a parabolic trajectory. The time it takes to reach the maximum height is equal to the time it takes for the ball to descend from the maximum height back to the initial level.
In this scenario, the time it takes for the ball to reach the maximum height and descend back is 0.910 seconds. Therefore, the total time for the ball to hit the ground is twice this value, which is 2 * 0.910 seconds = 1.820 seconds.
Next, we can calculate the vertical distance the ball traveled during this time using the equation of motion:
s = ut + 0.5 * gt^2
where:
s = vertical displacement (height of the cliff)
u = initial velocity (which is 0 since the ball starts from rest at the maximum height)
g = acceleration due to gravity (-32.2 ft/s^2)
t = time (1.820 seconds)
Plugging in the values, we get:
s = 0 * 1.820 + 0.5 * (-32.2) * (1.820)^2
s = 0 - 0.5 * 32.2 * 3.3124
s = -16.1 * 3.3124
s = -53.1644
The negative sign indicates that the displacement is in the opposite direction of the positive vertical axis. However, in this case, we are only interested in the magnitude of the displacement, so we ignore the negative sign.
Therefore, the vertical distance the ball traveled is approximately 53.1644 ft.
Since Bob was initially 5 ft above the edge of the cliff, the height of the cliff is 53.1644 ft + 5 ft = 58.1644 ft.
Thus, Bob climbed approximately 58.1644 feet.