Roger (mass of 75 kg) is travelling on a new ride called the Riddler's Revenge.
On a 7.1 m hill he is moving along at 31.3 m/s, then slows down to 24.7 m/s on the second hill.
How high is the second hill?
To determine the height of the second hill, we can use the principle of conservation of mechanical energy. The total mechanical energy at the first hill will be equal to the total mechanical energy at the second hill, assuming no energy losses due to friction or other factors.
The total mechanical energy of an object can be calculated by summing its kinetic energy (KE) and potential energy (PE). The kinetic energy of an object is given by the formula KE = 0.5 * mass * velocity^2, and the potential energy is given by PE = mass * gravity * height. In this case, the mass of Roger is 75 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
At the first hill:
Initial velocity (v1) = 31.3 m/s
Final velocity (v2) = 0 m/s (the object is momentarily at rest at the peak of the hill)
Using the equation KE1 + PE1 = KE2 + PE2, we can write:
0.5 * m * v1^2 + m * g * h1 = 0.5 * m * v2^2 + m * g * h2
Simplifying the equation, we get:
0.5 * m * (v1^2 - v2^2) = m * g * (h2 - h1)
Canceling out the mass, equation becomes:
0.5 * (v1^2 - v2^2) = g * (h2 - h1)
Now we can substitute the given values:
v1 = 31.3 m/s
v2 = 24.7 m/s
g = 9.8 m/s^2
h1 = 7.1 m (height of the first hill)
Plugging in these values:
0.5 * (31.3^2 - 24.7^2) = 9.8 * (h2 - 7.1)
Solving for h2:
h2 - 7.1 = (31.3^2 - 24.7^2) / (2 * 9.8)
h2 = (31.3^2 - 24.7^2) / (2 * 9.8) + 7.1
Calculating the value:
h2 ≈ 56.19 m
Therefore, the height of the second hill is approximately 56.19 meters.