a daredevil on a motorcycle leaves the ramp with a speed of 35.0 miles per hour. if his speed is 33.0 miles per hour when he reaches the peak of his path, what is the maximum height that he reaches. ignore friction and air resistance
His "launch angle" is
theta = cos^-1(33/35) = 19.46 degrees
The initial vertical velocity component is
Voy = 35*sin(19.46)
= 11.66 mph
= 17.1 ft/s
The time to reach maximum height is
t' = (Voy)/g = 0.53 seconds
Maximum height = (1/2)*Voy*t' = 4.5 feet PLUS the initial ramp height where he leaves it. They should have told you what that is.
Well, if we ignore friction and air resistance, we might as well ignore the laws of physics altogether, right?
But hey, I'm all about trying to crunch some imaginary numbers for you! So, let's get this circus started.
First, we'll need to convert those speeds to meters per second, because that's what the cool kids use. So, 35.0 mph is roughly 15.65 m/s, and 33.0 mph is around 14.75 m/s.
Now, remember that the maximum height is reached when the motorcycle's speed is at its minimum. According to our fantasy physics, the kinetic energy at the bottom of the ramp is equal to the potential energy at the peak. So, we can use the formula:
m * g * h = (1/2) * m * v^2
Where:
m = mass (which we conveniently forgot to mention)
g = acceleration due to gravity (around 9.8 m/s^2, but who's counting?)
h = maximum height
v = velocity at the peak
Since the mass and acceleration due to gravity will cancel each other out, we can simplify the equation to:
h = (1/2) * v^2
Plugging in the velocity of 14.75 m/s, we get:
h ≈ (1/2) * (14.75)^2
If you do the math, you should get the maximum imaginary height the daredevil reaches. Just be careful not to bump your head on the ceiling of reality while you're at it!
To find the maximum height reached by the daredevil, we can use the principle of conservation of energy. At the highest point of the motorcycle's path, all of the initial kinetic energy will be converted into potential energy.
Step 1: Convert the given speeds from miles per hour to meters per second.
We know that 1 mile per hour equals 0.44704 meters per second.
Given speed at the ramp = 35.0 miles per hour = 35.0 * 0.44704 meters per second ≈ 15.6468 meters per second.
Speed at the peak = 33.0 miles per hour = 33.0 * 0.44704 meters per second ≈ 14.7632 meters per second.
Step 2: Calculate the change in kinetic energy.
The change in kinetic energy is given by the formula: ΔKE = KE_f - KE_i, where KE_f is the final kinetic energy and KE_i is the initial kinetic energy.
Initial kinetic energy (KE_i) = 0.5 * mass * speed^2.
Final kinetic energy (KE_f) = 0.5 * mass * speed^2.
Since the mass is the same on the ramp and at the peak, we can cancel it out in the equation.
ΔKE = KE_f - KE_i = 0.5 * speed_f^2 - 0.5 * speed_i^2.
ΔKE = 0.5 * (14.7632^2 - 15.6468^2).
Step 3: Calculate the change in potential energy.
Using the principle of conservation of energy, ΔPE = -ΔKE.
ΔPE = -ΔKE ≈ -0.5 * (14.7632^2 - 15.6468^2).
Step 4: Calculate the maximum height reached.
Potential energy (PE) is given by the formula: PE = mass * gravitational acceleration * height.
ΔPE = PE_f - PE_i, where PE_f is the final potential energy, PE_i is the initial potential energy, and ΔPE is the change in potential energy.
PE_i = mass * gravitational acceleration * height_i = 0.
PE_f = mass * gravitational acceleration * height_f.
ΔPE = PE_f - PE_i = mass * gravitational acceleration * height_f - 0.
Therefore, mass * gravitational acceleration * height_f = ΔPE.
height_f = ΔPE / (mass * gravitational acceleration).
Substituting the value of ΔPE into the equation, we get:
height_f ≈ -ΔPE / (mass * gravitational acceleration).
height_f ≈ -(-0.5 * (14.7632^2 - 15.6468^2)) / (mass * gravitational acceleration).
Step 5: Substitute the known values and calculate the maximum height.
The mass of the daredevil and the gravitational acceleration are not given in the question. Let's assume the mass to be 100 kg and the acceleration due to gravity to be 9.8 m/s^2.
height_f ≈ -(-0.5 * (14.7632^2 - 15.6468^2)) / (100 * 9.8).
Using a calculator, we can find:
height_f ≈ 0.0305 meters or 30.5 centimeters.
Therefore, the maximum height reached by the daredevil on the motorcycle is approximately 0.0305 meters or 30.5 centimeters.
To find the maximum height reached by the daredevil, we can use the principles of mechanical energy conservation. At the highest point of the daredevil's path, all of his initial kinetic energy will have been converted into gravitational potential energy.
First, let's convert the given speeds from miles per hour to meters per second, as the metric system is commonly used in physics calculations.
1 mile = 1609 meters
1 hour = 3600 seconds
So, 35.0 miles per hour is approximately 15.65 meters per second, and 33.0 miles per hour is approximately 14.75 meters per second.
Now, let's denote the maximum height as h. At the highest point, the daredevil's speed is 33.0 miles per hour or 14.75 meters per second.
Using the principle of energy conservation, we can equate the initial kinetic energy to the final gravitational potential energy:
(1/2) * m * v_initial^2 = m * g * h
where m is the mass of the daredevil and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since the mass of the daredevil cancels out in both sides of the equation, we can simplify it to:
(1/2) * v_initial^2 = g * h
Plugging in the values, the equation becomes:
(1/2) * 15.65^2 = 9.8 * h
Simplifying further, we have:
122.7025 = 9.8 * h
To find the value of h, we divide both sides of the equation by 9.8:
h = 122.7025 / 9.8
Calculating this, we find:
h ≈ 12.52 meters
Therefore, the daredevil reaches a maximum height of approximately 12.52 meters.