A student is skateboarding down a ramp that is 5.79 m long and inclined at 17.6° with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp is 4.22 m/s. Neglect friction and find the speed at the bottom of the ramp.
To find the speed of the skateboarder at the bottom of the ramp, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of a system remains constant if no external forces, such as friction, are acting on it.
1. First, let's calculate the gravitational potential energy of the skateboarder at the top of the ramp.
The formula for gravitational potential energy is given by: PE = m * g * h, where
- PE is the potential energy (in Joules),
- m is the mass of the skateboarder (which is not given),
- g is the acceleration due to gravity (which is approximately 9.8 m/s²),
- h is the height of the skateboarder above the bottom of the ramp.
Since the ramp is inclined at an angle of 17.6°, we can use trigonometry to find the height of the skateboarder.
h = sin(17.6°) * 5.79 m
2. Next, we calculate the kinetic energy of the skateboarder at the top of the ramp.
The formula for kinetic energy is given by: KE = (1/2) * m * v², where
- KE is the kinetic energy (in Joules),
- m is the mass of the skateboarder (which is not given),
- v is the initial speed of the skateboarder at the top of the ramp (4.22 m/s).
3. Since we are neglecting friction, the total mechanical energy of the system remains constant.
Therefore, the initial mechanical energy (Ei) at the top of the ramp is equal to the final mechanical energy (Ef) at the bottom of the ramp.
Ei = Ef
4. The initial mechanical energy is the sum of the gravitational potential energy and the kinetic energy at the top of the ramp. Thus:
Ei = PE_initial + KE_initial
5. The final mechanical energy is the sum of the gravitational potential energy and the kinetic energy at the bottom of the ramp. Thus:
Ef = PE_final + KE_final
6. Since the ramp is horizontal at the bottom, the gravitational potential energy is zero. So:
Ef = PE_final + KE_final = 0 + KE_final = KE_final
7. Equating the initial and the final mechanical energies (from steps 3 and 4):
Ei = Ef
PE_initial + KE_initial = KE_final
8. Substituting the equations from steps 1 and 2 into step 7:
(m * g * h) + (1/2) * m * v² = KE_final
9. Now, we can solve for the final kinetic energy (KE_final):
KE_final = (m * g * h) + (1/2) * m * v²
10. Finally, to find the speed at the bottom of the ramp, we can use the formula for kinetic energy:
KE_final = (1/2) * m * (v_final)²
11. Rearranging the equation from step 10 to solve for the speed (v_final):
v_final = √[(2 * KE_final) / m]
Please note that the mass of the skateboarder is not provided in the question. To obtain the actual numerical answer, you would need to know the skateboarder's mass.