Divers in Acapulco dive from a cliff that is 56 m high. If the rocks below the cliff extend outward for 23 m, what is the minimum horizontal velocity a diver must have to clear the rocks?
Vx > 23 m/(fall time from cliff)
fall time = sqrt (2H/g)= 3.38 s
The data are phoney here. The divers would have to be running at a 4-minute-mile rate towards the cliff before they jump. The actual cliff heght is 30 meters and I do not believe the cliff extends out as far as 23 m at the place where they land. I have seen this dive performed and can verify that the numbers are wrong.
To determine the minimum horizontal velocity a diver must have to clear the rocks, we need to consider the vertical motion and the horizontal motion separately. Let's break down the problem step by step:
1. Start by understanding the motion of the diver in the vertical direction:
The cliff is 56 meters high. Using the principles of projectile motion, we can determine the time it takes for the diver to reach the maximum height and then fall to the rocks below.
a. The time to reach maximum height can be calculated using the equation v = u + at, where v is the final vertical velocity, u is the initial vertical velocity, a is the acceleration, and t is the time.
- Since the diver jumps from rest at the top of the cliff, the initial vertical velocity (u) is 0 m/s.
- The final vertical velocity (v) at the maximum height is also 0 m/s since the diver momentarily stops before falling.
- The acceleration (a) is the acceleration due to gravity, approximately 9.8 m/s² (assuming negligible air resistance).
- Solving the equation v = u + at for t, we get t = v/a = 0/9.8 = 0 seconds.
b. To determine the time it takes for the diver to fall from the maximum height to the rocks below, we can use the equation d = ut + (1/2)at², where d is the vertical distance traveled.
- The initial velocity (u) is 0 m/s since the diver starts falling from rest at the top of the cliff.
- The acceleration (a) is the same as before, 9.8 m/s².
- The vertical distance traveled (d) is the height of the cliff minus the extension of the rocks, so d = 56 m - 23 m = 33 m.
- Solving the equation d = ut + (1/2)at² for t, we get t = √(2d/a) = √(2*33/9.8) ≈ 2.02 seconds.
2. Next, determine the horizontal distance the diver needs to clear:
Since we know the time (t) it takes for the diver to fall, we can now find the horizontal distance (x) by using the formula x = vt, where v is the horizontal velocity.
- We need to solve for v, so rearrange the equation as v = x/t.
- The required horizontal distance to clear the rocks is given as the distance the rocks extend outward, which is 23 meters.
- The time (t) is the time it takes for the diver to fall, which we calculated earlier as 2.02 seconds.
- Plugging in the values, we find v = 23 m / 2.02 s ≈ 11.39 m/s.
Therefore, the minimum horizontal velocity a diver must have to clear the rocks is approximately 11.39 m/s.