A student is skateboarding down a ramp that is 6.03 m long and inclined at 21.5° with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp is 4.87 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 energy.
First, let's determine the potential energy at the top and the kinetic energy at the bottom of the ramp.
1. Potential Energy (PE) at the top of the ramp:
The potential energy formula is given by: PE = m * g * h,
where m is the mass of the skateboarder, g is the acceleration due to gravity (9.8 m/s²), and h is the vertical distance from the bottom of the ramp to the top.
In this case, the vertical distance (h) is given by h = ramp length * sin(angle) = 6.03 m * sin(21.5°).
Therefore, the potential energy at the top of the ramp is PE_top = m * g * h.
2. Kinetic Energy (KE) at the bottom of the ramp:
The kinetic energy formula is given by: KE = 0.5 * m * v²,
where v is the velocity of the skateboarder at the bottom of the ramp.
Therefore, the kinetic energy at the bottom of the ramp is KE_bottom = 0.5 * m * v².
According to the conservation of energy, the total mechanical energy remains constant, thus:
PE_top + KE_top = PE_bottom + KE_bottom.
Since we are neglecting friction, the only energy change is due to the change in potential and kinetic energy of the skateboarder.
3. Setting up the conservation of energy equation:
PE_top + KE_top = PE_bottom + KE_bottom
m * g * h + 0.5 * m * v_top² = 0 + 0.5 * m * v_bottom²
Note that the potential energy at the bottom is zero since we set the reference point at the bottom of the ramp.
4. Simplifying the equation:
m * g * h = 0.5 * m * v_bottom²
g * h = 0.5 * v_bottom²
5. Solving for v_bottom:
v_bottom² = 2 * g * h
v_bottom = √(2 * g * h)
Now, substitute the given values into the equation to find v_bottom:
g = 9.8 m/s² (acceleration due to gravity)
h = 6.03 m * sin(21.5°) (vertical distance)
Plug in these values and calculate the final velocity v_bottom.
h = 6.03*sin21.5 = 2.21 m.
V^2 = Vo^2 + 2g*h = 4.87^2 + 19.6*2.21
V = Final velocity
Vo = 4.87 m/s
g = 9.8 m/s^2
Solve for V.