A student is skateboarding down a ramp that is 6.52 m long and inclined at 19.8 degrees with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp 2.24 m/s. Neglect friction and find the speed at the bottom of the ramp in m/s.
To find the speed at the bottom of the ramp, we can use the principle of conservation of energy. The potential energy at the top of the ramp is converted into kinetic energy at the bottom.
Step 1: Find the initial potential energy at the top of the ramp.
The potential energy (PE) is given by the formula:
PE = mass * gravity * height
Since the mass is not given, we can cancel it out by dividing both sides of the equation by the mass:
PE/mass = gravity * height
Step 2: Find the height of the ramp.
The height can be found using trigonometry. The vertical height (h) is given by the equation:
h = ramp length * sin(ramp angle)
Substituting the given values:
h = 6.52 m * sin(19.8 degrees)
Step 3: Calculate the initial potential energy.
Using the equation from step 1:
PE = gravity * height
= 9.8 m/s^2 * (6.52 m * sin(19.8 degrees))
Step 4: Find the final kinetic energy at the bottom of the ramp.
The kinetic energy (KE) is given by the formula:
KE = 1/2 * mass * velocity^2
Since we want to find the final velocity (speed), we can rearrange the equation:
KE = 1/2 * mass * velocity^2
velocity = sqrt(2 * KE / mass)
Step 5: Calculate the final kinetic energy.
Using the conservation of energy, the initial potential energy is equal to the final kinetic energy:
PE = KE
gravity * height = 1/2 * mass * velocity^2
Step 6: Solve for the final velocity.
Plugging in the known values:
gravity * height = 1/2 * mass * velocity^2
9.8 m/s^2 * (6.52 m * sin(19.8 degrees)) = 1/2 * mass * velocity^2
velocity = sqrt((2 * 9.8 m/s^2 * (6.52 m * sin(19.8 degrees)) / mass)
Step 7: Calculate the value of the final velocity.
Substituting the known values into the equation:
velocity = sqrt((2 * 9.8 m/s^2 * (6.52 m * sin(19.8 degrees)) / mass)
= sqrt((2 * 9.8 m/s^2 * (6.52 m * sin(19.8 degrees)) / mass)
Therefore, the speed at the bottom of the ramp will be equal to sqrt((2 * 9.8 m/s^2 * (6.52 m * sin(19.8 degrees)) / mass).
To find the speed at the bottom of the ramp, we can use the principles of conservation of energy. At the top of the ramp, the skateboarder has both kinetic energy and potential energy. At the bottom of the ramp, all the potential energy has been converted to kinetic energy.
To calculate the speed at the bottom of the ramp, we will first calculate the potential energy at the top and then convert it to kinetic energy at the bottom.
Step 1: Calculate the initial potential energy at the top of the ramp.
Potential energy (PE) can be calculated using the equation:
PE = m * g * h
Where m is the mass of the skateboarder, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the vertical height from the top of the ramp to the bottom (6.52 * sin(19.8)).
Step 2: Calculate the initial kinetic energy at the top of the ramp.
Kinetic energy (KE) can be calculated using the equation:
KE = 0.5 * m * v^2
Where m is the mass of the skateboarder (which we can assume to be constant), and v is the initial speed of the skateboarder (2.24 m/s).
Step 3: Set the initial potential energy equal to the final kinetic energy and solve for the final speed.
PE = KE
m * g * h = 0.5 * m * v^2
Canceling out the mass on both sides of the equation, we get:
g * h = 0.5 * v^2
Now, we can solve for v by rearranging the equation:
v^2 = 2 * g * h
v = sqrt(2 * 9.8 * (6.52 * sin(19.8)))
Finally, we can calculate the speed at the bottom of the ramp.
v = sqrt(2 * 9.8 * 6.52 * sin(19.8))
v ≈ 5.94 m/s
Therefore, the speed at the bottom of the ramp is approximately 5.94 m/s.
h = 6.52/sin 19.8
loss of potential E = m g h
(1/2) m v^2 = m g h + (1/2) m (2.24)^2
m cancels
v^2 = 2 g h + 2.24^2