A daredevil on a motorcycle leaves the end of a ramp with a speed of 33.0 m/s as in the figure below. If his speed is 31.3 m/s when he reaches the peak of the path, what is the maximum height that he reaches? Ignore friction and air resistance.

Since the horizontal component of his speed stays the same (31.3 m/s in this case), the vertical component (which is zero at maximum height) must initially be sqrt(33.0^2 - 31.3^2) = 10.46 m/s. The time spent going up is

T = (10.46 m/s)/g = 1.07 s

The height that it reaches is
H = Vy,average* T
= (Vy,initial/2)*T = 5.8 meters

To find the maximum height reached by the daredevil, we can use the principle of conservation of mechanical energy. At the peak of the path, the daredevil's potential energy (due to height) will be at its maximum, while his kinetic energy (due to motion) will be zero.

1. We can start by calculating the initial kinetic energy (KEi) of the daredevil at the end of the ramp using the formula:
KEi = 0.5 * m * v^2, where m is the mass of the daredevil and v is his velocity.
Since we are not given the mass of the daredevil, we can assume a value of 1 kg in this case.

KEi = 0.5 * 1 kg * (33.0 m/s)^2
= 544.5 J (rounded to one decimal place)

2. At the peak of the path, the daredevil's kinetic energy will be zero. However, his potential energy (PE) will be at its maximum. The formula for potential energy is:
PE = m * g * h, where g is the acceleration due to gravity (9.8 m/s^2) and h is the maximum height reached.

PE = 1 kg * 9.8 m/s^2 * h
Since the daredevil has zero kinetic energy at the peak, all of his initial kinetic energy will be converted into potential energy.

3. Equating the initial kinetic energy (KEi) to the potential energy (PE) at the peak, we get:
KEi = PE
544.5 J = 1 kg * 9.8 m/s^2 * h
h = 544.5 J / (1 kg * 9.8 m/s^2)
h ≈ 55.6 meters (rounded to two decimal places)

Therefore, the daredevil reaches a maximum height of approximately 55.6 meters.