A 16kg child on roller skates rolls 2m down an incline at 20 degrees with the horizontal. With no friction, what is the total kinetic energy of the child at the bottom of the incline?
It falls a vertical distance H = 2 sin 20 meters.
In the process, the kinetic energy increases by M g H.
To calculate the total kinetic energy of the child at the bottom of the incline, we need to consider the gravitational potential energy the child loses as they move down the incline and the corresponding increase in kinetic energy.
The formula for gravitational potential energy is given by:
Potential Energy = mass * gravitational acceleration * height
In this case, the height is the vertical component of the 2-meter incline, which can be found by multiplying the length of the incline by the sine of the incline angle:
Height = length of incline * sin(angle)
The mass of the child is given as 16 kg, and the gravitational acceleration is approximately 9.8 m/s^2.
With these values, we can calculate the potential energy:
Potential Energy = 16 kg * 9.8 m/s^2 * (2 m * sin(20 degrees))
To find the kinetic energy at the bottom of the incline, we use the fact that the total mechanical energy is conserved, meaning the potential energy lost is converted into kinetic energy gained:
Kinetic Energy = Potential Energy
Therefore, the total kinetic energy of the child at the bottom of the incline is equal to the calculated potential energy.
To calculate the total kinetic energy of the child at the bottom of the incline, we need to first calculate the gravitational potential energy at the top of the incline, and then convert it into kinetic energy at the bottom.
1. Calculate the gravitational potential energy at the top of the incline:
- The formula to calculate gravitational potential energy is given by: PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height or vertical distance.
- The mass of the child is given as 16 kg.
- The acceleration due to gravity (g) is approximately 9.8 m/s^2.
- The height or vertical distance is given by the vertical component of the 2m distance, which can be calculated as h = 2m * sin(20°).
- Substitute the values into the formula:
PE = 16 kg * 9.8 m/s^2 * (2 m * sin(20°)).
2. Convert the gravitational potential energy into kinetic energy at the bottom:
- The total mechanical energy (conservation of energy) remains constant without friction, hence the potential energy at the top is converted into kinetic energy at the bottom.
- Kinetic energy is given by the formula: KE = 1/2 * m * v^2, where m is the mass and v is the velocity.
- We need to find the velocity of the child at the bottom of the incline.
- The horizontal component of the distance is given by: d = 2m * cos(20°).
- The time taken to cover this horizontal distance can be calculated using the equation: d = v * t, where d is the distance and t is the time.
- Substitute the values into the equation:
2m * cos(20°) = v * t.
- Solve for t: t = (2m * cos(20°)) / v.
- We know that the vertical distance (h) can be replaced by the initial vertical velocity (v₀) using the equation: h = v₀ * t + 1/2 * g * t^2.
- By substituting the values, we can solve for v₀. Since the child starts from rest at the top, v₀ becomes zero.
- Substitute the initial velocity (v₀ = 0) and the time (t) into the equation to solve for the final velocity (v).
- Now substituting the mass (16 kg) and the velocity (v) into the formula for kinetic energy:
KE = 1/2 * 16 kg * v^2.
So, in summary, the total kinetic energy of the child at the bottom of the incline is given by the formula KE = 1/2 * 16 kg * v^2, where v is the final calculated velocity.