A student is skateboarding down a ramp that is 5.38 m long and inclined at 19.4 degrees with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp 2.52 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 mechanical energy. The mechanical energy at the top of the ramp (due to the gravitational potential energy) will be converted to kinetic energy at the bottom of the ramp.
The gravitational potential energy at the top of the ramp can be calculated using the equation:
PE = m * g * h
Where PE is the potential energy, m is the mass of the skateboarder, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the ramp.
The height of the ramp can be found using trigonometry. The vertical height of the ramp (h) can be calculated using the equation:
h = L * sin(theta)
Where L is the length of the ramp and theta is the angle of inclination.
Now, let's plug in the given values:
L = 5.38 m
theta = 19.4 degrees
v_initial = 2.52 m/s
g = 9.8 m/s^2
First, convert the angle from degrees to radians:
theta_radians = theta * (pi/180)
Calculate the height of the ramp:
h = L * sin(theta_radians)
Now, we can find the potential energy at the top of the ramp:
PE = m * g * h
Next, we can equate the potential energy at the top of the ramp to the kinetic energy at the bottom of the ramp:
PE = KE
The kinetic energy can be calculated using the equation:
KE = 0.5 * m * v^2
Where v is the velocity at the bottom of the ramp.
Solving for v, we have:
v^2 = (2 * PE) / m
Finally, take the square root of both sides to find the velocity at the bottom of the ramp.