A daredevil on a motorcycle leaves the end of a ramp with a speed of 30.5 m/s as in the figure below. If his speed is 29.0 m/s when he reaches the peak of the path, what is the maximum height that he reaches? Ignore friction and air resistance.
m
Diagram?
To find the maximum height that the daredevil reaches, we can use the principles of conservation of mechanical energy. At the start of the jump, the motorcycle has a certain amount of kinetic energy due to its initial speed. At the peak of the path, all of this kinetic energy is converted into potential energy, since the motorcycle momentarily comes to a stop before descending.
The law of conservation of mechanical energy states that the total mechanical energy of a system remains constant, assuming no external forces are acting on it. Mathematically, this can be expressed as:
Initial Kinetic Energy + Initial Potential Energy = Final Kinetic Energy + Final Potential Energy
In this case, we can assume that the motorcycle starts at ground level, so its initial potential energy is zero. Therefore, the equation becomes:
(1/2) * m * initial_velocity^2 = m * g * height
where:
m = mass of the daredevil and the motorcycle
initial_velocity = initial speed of the motorcycle at the end of the ramp
g = acceleration due to gravity
height = maximum height reached by the motorcycle
Plugging in the given values:
(1/2) * m * (30.5 m/s)^2 = m * 9.8 m/s^2 * height
We can cancel out the mass 'm' on both sides of the equation, leaving us with:
(1/2) * (30.5 m/s)^2 = 9.8 m/s^2 * height
Simplifying further:
height = [(1/2) * (30.5 m/s)^2] / (9.8 m/s^2)
height ≈ 46.925 meters
Therefore, the maximum height reached by the daredevil on the motorcycle is approximately 46.925 meters.