A student is skateboarding down a ramp that is 5.9 m long and inclined at 14° with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp is 2.5 m/s. Neglect friction and find the speed at the bottom of the ramp.
h = 5.9*sin14 = 1.43 m. = ht. of ramp.
V^2 = Vo^2 + 2g*h.
V^2 = 2.5 + 19.6*1.43 = 30.53,
V = 5.53 m/s.
To find the speed of the skateboarder at the bottom of the ramp, we can use the principles of conservation of energy. The initial gravitational potential energy at the top of the ramp is converted into kinetic energy at the bottom.
1. Calculate the initial potential energy at the top of the ramp:
Potential energy (PE) = mass (m) * gravitational constant (g) * height (h)
Since the ramp is inclined at an angle of 14°, the height can be calculated as:
height = length of the ramp (5.9 m) * sin(angle)
PE = m * g * height
2. Calculate the initial potential energy at the top of the ramp in terms of kinetic energy at the bottom:
PE = (1/2) * m * v^2
where v is the velocity at the bottom of the ramp.
3. Set the equations equal to each other and solve for v:
(1/2) * m * v^2 = m * g * height
Simplify and solve for v:
v^2 = 2 * g * height
v = √(2 * g * height)
where g is the acceleration due to gravity (9.8 m/s^2) and height is calculated in step 1.
Substituting the values given:
height = 5.9 m * sin(14°)
height = 1.576 m
v = √(2 * 9.8 * 1.576)
v ≈ 3.90 m/s
Therefore, the skateboarder's speed at the bottom of the ramp is approximately 3.90 m/s.