A daredevil on a motorcycle leaves the end of a ramp with a speed of 35.5 m/s as in the figure below. If his speed is 34.0 m/s when he reaches the peak of the path, what is the maximum height that he reaches? Ignore friction and air resistance.
m
since speed at the top is all horizontal, we know that
35.5 cosθ = 34
Now you can figure the vertical speed at takeoff, so use what you know about max height given initial vertical speed.
To find the maximum height reached by the daredevil, we can use the principle of conservation of energy. At the top of the path, the kinetic energy of the daredevil's motorcycle is zero, and all the initial energy is converted into potential energy.
The total mechanical energy (E) of the system at the start and at the top of the path remains constant. It is given by the sum of kinetic energy (KE) and potential energy (PE):
E = KE + PE
At the top of the path, the kinetic energy is zero, and thus E = PE.
The potential energy (PE) is given by the equation:
PE = mgh
Where:
m = mass of the daredevil and the motorcycle (given)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height (maximum height reached)
Since the daredevil is on a ramp, we can assume that the height he reaches is the vertical distance from the starting point to the top of the ramp.
To find the maximum height, we need to rearrange the equation to solve for h:
h = PE / (mg)
Substituting the known values:
h = E / (mg)
First, let's calculate the initial mechanical energy (E) of the system at the start of the ramp:
E = KE + PE
Since friction and air resistance are ignored, the only form of energy at the start is kinetic energy.
KE = 1/2 mv^2
Substituting the given values:
KE = 1/2 * m * (35.5 m/s)^2
Calculate the value of KE.
Then, plug in the values of m, g, and the calculated value of KE into the equation for h:
h = E / (mg)
Finally, calculate the value of h.
Note: The figure mentioned in the question is not provided, but it is not necessary to solve the problem. If you have the figure, you can visualize the situation better, but the calculations can still be done without it.