A student is skateboarding down a ramp that is 6.88 m long and inclined at 16.9 degrees with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp 2.54 m/s. Neglect friction and find the speed at the bottom of the ramp in m/s.

Find the real height, rather than the slant height. Use Sin Cos or Tan. Plug that value into PE=mgh solve for PE.

PE=KE
KE=1/2mv^2
Solve for v

To find the speed at the bottom of the ramp, we can use the principle of conservation of energy. The total mechanical energy at the top of the ramp (which includes the kinetic energy and potential energy) will be equal to the total mechanical energy at the bottom of the ramp.

The initial mechanical energy at the top of the ramp is the sum of the kinetic energy (KE) and the potential energy (PE), given by:

Initial Mechanical Energy (at top) = KE + PE

The final mechanical energy at the bottom of the ramp is the kinetic energy (KE) only, given by:

Final Mechanical Energy (at bottom) = KE

According to the principle of conservation of energy:

Initial Mechanical Energy (at top) = Final Mechanical Energy (at bottom)

Let's calculate the initial mechanical energy (at top) first:

Initial Kinetic Energy (at top) = (1/2) * mass * (initial velocity)^2

Given that the initial speed is 2.54 m/s, we can square it to get the initial velocity.

Initial Kinetic Energy (at top) = (1/2) * mass * (2.54 m/s)^2

Next, we calculate the initial potential energy (at top):

Initial Potential Energy (at top) = mass * acceleration due to gravity * height

Since the ramp is inclined at an angle with respect to the horizontal, we need to find the vertical height.

Vertical Height = ramp length * sin(ramp angle)

Given that the ramp length is 6.88 m and the ramp angle is 16.9 degrees, we can calculate the vertical height.

Vertical Height = 6.88 m * sin(16.9 degrees)

Now, we can calculate the initial potential energy (at top):

Initial Potential Energy (at top) = mass * acceleration due to gravity * vertical height

Finally, we can calculate the total initial mechanical energy (at top) by adding the initial kinetic energy and the initial potential energy:

Initial Mechanical Energy (at top) = Initial Kinetic Energy (at top) + Initial Potential Energy (at top)

Now that we have the initial mechanical energy (at top), we can equate it to the final mechanical energy (at bottom) to find the final kinetic energy (at bottom):

Final Mechanical Energy (at bottom) = Initial Mechanical Energy (at top)

Since the final mechanical energy (at bottom) is the final kinetic energy (at bottom), we can solve for the final kinetic energy (at bottom):

Final Kinetic Energy (at bottom) = Final Mechanical Energy (at bottom)

The final kinetic energy (at bottom) can be expressed as:

Final Kinetic Energy (at bottom) = (1/2) * mass * (final velocity)^2

Now, we know that the final mechanical energy (at bottom) is equal to the final kinetic energy (at bottom), so we can equate the two expressions:

(1/2) * mass * (final velocity)^2 = Final Mechanical Energy (at bottom)

Now, we can solve for the final velocity (at bottom) by rearranging the equation:

(final velocity)^2 = 2 * Final Mechanical Energy (at bottom) / mass

Finally, we can take the square root of both sides to find the final velocity (at bottom):

final velocity (at bottom) = √(2 * Final Mechanical Energy (at bottom) / mass)

By substituting the known values and calculating the final velocity, we will get the speed at the bottom of the ramp in m/s.