A daredevil on a motorcycle leaves the end of a ramp with a speed of 34.5 m/s as in the figure below. If his speed is 32.1 m/s when he reaches the peak of the path, what is the maximum height that he reaches? Ignore friction and air resistance.
well, if his horzontal velocity is 32.1m/s, and his total at launch is 34.5m/s,Then his initial vertical velocity is sqrt (34.5^2-32.1^2)=12.6m/s check that.
Vi^2=2gh
solve for h.
The 32.11 m/s is the unchanging horizontal component. That and Mr. Pythagoras will tell you the initial vertical component.
sqrt[(35.5)^2 - (32.1)^2] = Vyo
From Vyo, you can derive the maximum height.
Go for it!
To calculate the maximum height reached by the daredevil, we can use the principle of conservation of mechanical energy.
The principle of conservation of mechanical energy states that the total mechanical energy (sum of kinetic and potential energy) of an object remains constant in the absence of non-conservative forces like friction and air resistance.
At the bottom of the ramp, the mechanical energy is entirely in the form of kinetic energy:
KE1 = (1/2) * m * v1^2, where m is the mass of the daredevil and v1 is the speed at the bottom of the ramp.
At the top of the path, the mechanical energy is entirely in the form of potential energy:
PE2 = m * g * h, where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the maximum height reached by the daredevil.
Since the total mechanical energy is conserved, we can equate the initial kinetic energy to the final potential energy:
KE1 = PE2
(1/2) * m * v1^2 = m * g * h
We can rearrange the formula to solve for h:
h = (1/2) * v1^2 / g
Substituting the given values:
v1 = 32.1 m/s
g = 9.8 m/s^2
h = (1/2) * (32.1 m/s)^2 / 9.8 m/s^2
Calculating the result:
h = (1/2) * (1030.41 m^2/s^2) / 9.8 m/s^2
h = 528.24 m
Therefore, the maximum height reached by the daredevil is approximately 528.24 meters.
To find the maximum height that the daredevil reaches, we can use the concept of conservation of energy. At the peak of the path, the kinetic energy of the daredevil is completely converted into potential energy.
The initial kinetic energy of the daredevil can be calculated using the formula:
K.E. = (1/2) * m * v^2,
where m is the mass of the daredevil (which we can assume to be given or negligible) and v is the velocity at the end of the ramp.
Plugging in the given values, we have:
K.E. = (1/2) * v^2.
Since energy is conserved, the potential energy at the peak of the path is equal to the initial kinetic energy. The potential energy can be calculated using the formula:
P.E. = m * g * h,
where m is the mass of the daredevil, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum height.
Setting the initial kinetic energy equal to the potential energy, we have:
(1/2) * v^2 = m * g * h.
Simplifying and solving for h, we get:
h = (1/2) * (v^2) / (m * g).
Since the mass of the daredevil is not given, we can cancel it out and the equation becomes:
h = (1/2) * (v^2) / g.
Now, let's plug in the given values:
h = (1/2) * (32.1 m/s)^2 / 9.8 m/s^2.
Calculating this expression, the maximum height is approximately 52.04 meters.
Therefore, the daredevil reaches a maximum height of approximately 52.04 meters.