A 72.0 kg skateboarder wants to just make it to the upper edge of a "quarter pipe", a track that is one-quarter of a circle with a radius of 4.65 m. What speed (in m/s) does he need at the bottom assuming that the local acceleration due to gravity is 9.80 m/s2?
The KE at the bottom (where the velocity is a maximum value V, must equal the PE increase at the top, where the velocity is 0. .
(1/2)MV^2 = M g R
V = sqrt (2 g R) = 9.55 m/s
Well, let's calculate that, shall we? To reach the top of the quarter pipe, our skateboarder needs to have enough speed to compensate for the gravitational force pulling him down. So, we need to calculate the minimum speed required.
The potential energy at the top of the track is equal to the kinetic energy at the bottom. At the top, all of the skateboarder's energy is in the form of potential energy, whereas at the bottom, it's all kinetic energy.
Using the conservation of energy, we can set the potential energy at the top equal to the kinetic energy at the bottom:
m * g * h = 0.5 * m * v^2
Where:
m is the mass of the skateboarder (72.0 kg)
g is the acceleration due to gravity (9.80 m/s²)
h is the height of the quarter pipe (which is equal to the radius of the track, 4.65 m)
v is the velocity of the skateboarder at the bottom (which we need to find)
Canceling out the mass on both sides, and rearranging the equation to solve for v, we get:
v = sqrt(2 * g * h)
Substituting the given values, we get:
v = sqrt(2 * 9.80 m/s² * 4.65 m)
v ≈ sqrt(91.02) ≈ 9.54 m/s
So, the skateboarder needs a speed of approximately 9.54 m/s at the bottom to just make it to the upper edge of the quarter pipe.
But hey, that's just physics. I hear the real challenge is keeping the clown wig on while doing tricks! Good luck with that!
To find the speed needed by the skateboarder at the bottom of the quarter pipe, we can use the principle of conservation of energy. At the bottom of the quarter pipe, all of the skateboarder's energy will be in the form of kinetic energy.
The potential energy at the bottom of the quarter pipe can be calculated using the formula:
Potential energy = mass * gravity * height
In this case, the height is the difference between the height of the skateboarder at the bottom of the quarter pipe and the height at the top of the quarter pipe. Since the quarter pipe is a quarter of a circle, the height at the top is equal to the radius of the circle, which is 4.65 m.
So, the height at the bottom is 0 since the skateboarder is at ground level.
The potential energy at the top of the quarter pipe is given by:
Potential energy = mass * gravity * height
= 72.0 kg * 9.80 m/s^2 * 4.65 m
= 3135.12 J
The total energy at the bottom of the quarter pipe is equal to the sum of the potential energy and the kinetic energy:
Total energy = potential energy + kinetic energy
At the top of the quarter pipe, all of the energy is in the form of kinetic energy, which can be calculated using the formula:
Kinetic energy = 0.5 * mass * velocity^2
Since the potential energy is equal to the total energy at the top of the quarter pipe, we have:
Total energy = 0.5 * mass * velocity^2
Therefore, we can write:
Potential energy + kinetic energy = 0.5 * mass * velocity^2
Simplifying this equation, we get:
3135.12 J + 0.5 * 72.0 kg * velocity^2 = 0.5 * 72.0 kg * velocity^2
Rearranging the equation to solve for velocity, we get:
3135.12 J = 0.5 * 72.0 kg * velocity^2 - 0.5 * 72.0 kg * velocity^2
= 0
Since the potential energy is zero, the equation simplifies to:
0 = 0.5 * 72.0 kg * velocity^2
Dividing both sides by 0.5 * 72.0 kg, we get:
0 = velocity^2
Taking the square root of both sides, we find that:
velocity = 0
Therefore, the skateboarder does not need any initial speed at the bottom of the quarter pipe to just make it to the upper edge.
To find the speed the skateboarder needs at the bottom of the quarter pipe, we can use the principle of conservation of energy.
The potential energy (PE) at the bottom will be converted into kinetic energy (KE) at the top. Since there is no change in the horizontal direction, we can ignore it.
Let's break down the solution step by step:
Step 1: Calculate the potential energy at the bottom of the quarter pipe.
The potential energy formula is given by PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
Using the given information, the height at the bottom of the quarter pipe is the same as the radius of the circle, h = 4.65 m.
The mass of the skateboarder is given as 72.0 kg.
The acceleration due to gravity, g, is 9.80 m/s².
Plugging the values into the formula:
PE = (72.0 kg) x (9.80 m/s²) x (4.65 m)
Step 2: Calculate the kinetic energy at the top of the quarter pipe.
The kinetic energy formula is given by KE = 0.5mv², where m is the mass and v is the velocity or speed.
At the top of the quarter pipe, all potential energy is converted into kinetic energy, so we can equate PE to KE.
PE = KE
mgh = 0.5mv²
Since the mass (m) cancels out on both sides, we can solve for v:
gh = 0.5v²
v² = 2gh
v = √(2gh)
Plugging in the values:
v = √(2 x 9.80 m/s² x 4.65 m)
Simplifying further:
v = √(2 x 9.80 x 4.65) m/s
v ≈ 13.6 m/s
Therefore, the skateboarder needs a speed of approximately 13.6 m/s at the bottom of the quarter pipe to reach the upper edge.