Find the height of a cliff if the 87.6 kg cliff diver hits the water at 15.5 m/s. Find the answer to this one using energy equations. Then find the answer using a motion equation.
To find the height of the cliff, you can use both energy equations and motion equations.
Using Energy Equations:
1. Start by calculating the potential energy (PE) of the cliff diver when they are at the starting point on the cliff. The potential energy formula is PE = mgh, where m is the mass of the cliff diver (87.6 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height of the cliff (unknown).
2. Next, calculate the kinetic energy (KE) of the cliff diver just before hitting the water using the formula KE = 1/2mv², where m is the mass of the cliff diver (87.6 kg) and v is the velocity of the diver just before hitting the water (15.5 m/s).
3. According to the law of conservation of energy, the total mechanical energy is conserved. Therefore, the initial potential energy (mgh) should be equal to the final kinetic energy (1/2mv²). So, set up the equation mgh = 1/2mv² and solve for h.
Using Motion Equations:
1. Determine the value of the acceleration due to gravity (g = 9.8 m/s²).
2. Use the kinematic equation s = ut + 1/2at², where s is the displacement (height of the cliff), u is the initial velocity (0 m/s), t is the time taken to fall, and a is the acceleration due to gravity (-9.8 m/s²).
3. Rearrange the equation to h = 1/2at² and solve for h.
Now, let's calculate the height of the cliff using both methods.
Using Energy Equations:
PE = mgh
KE = 1/2mv²
mgh = 1/2mv²
h = (1/2mv²) / (mg)
h = (1/2 * 87.6 kg * (15.5 m/s)²) / (87.6 kg * 9.8 m/s²)
h = (1/2 * 87.6 * 240.25) / (87.6 * 9.8)
h = 12012.9 / 857.28
h ≈ 14 meters (rounded to the nearest meter)
Using Motion Equations:
h = 1/2at²
h = 1/2 * 9.8 m/s² * t²
h = 4.9 m/s² * t²
We need to find the time it takes for the cliff diver to hit the water. We can use another kinematic equation to find this time.
v = u + at, where v = 15.5 m/s, u = 0 m/s, and a = 9.8 m/s² (acceleration due to gravity)
15.5 m/s = 0 m/s + 9.8 m/s² * t
15.5 m/s = 9.8 m/s² * t
t = 15.5 m/s / 9.8 m/s²
t ≈ 1.58 seconds (rounded to two decimal places)
Now, substitute the calculated time back into the motion equation:
h = 4.9 m/s² * (1.58 s)²
h = 4.9 m/s² * 2.4964 s²
h ≈ 12.22 meters (rounded to two decimal places)
Using motion equations, the height of the cliff is approximately 12.22 meters. Using energy equations, the height is approximately 14 meters.