A child and sled with a combined mass of 48.4 kg slide down a frictionless hill that is 7.15 m high. If the sled starts from rest, what is its speed at the bottom of the hill?
V^@=Vo^2 + 2g*h = 0 + 19.6*7.15 = 140.14
V = 11.84 m/s.
Correction: Change V^@ to V^2.
To find the sled's speed at the bottom of the hill, we can use the principle of conservation of mechanical energy. Initially, the sled is at rest, so it has no kinetic energy. Its total mechanical energy is equal to its potential energy at the top of the hill.
Here's how we can calculate the speed:
Step 1: Calculate the potential energy at the top of the hill.
Potential energy (PE) = mass (m) × gravitational acceleration (g) × height (h)
Given:
mass of the sled and child (m) = 48.4 kg
height of the hill (h) = 7.15 m
gravitational acceleration (g) = 9.8 m/s^2
PE = 48.4 kg × 9.8 m/s^2 × 7.15 m
Step 2: Calculate the kinetic energy at the bottom of the hill using the conservation of mechanical energy.
At the bottom of the hill, all the potential energy is converted into kinetic energy.
Kinetic energy (KE) = 1/2 × mass (m) × velocity^2
Since the sled starts from rest, its initial velocity (v) is 0.
KE = 1/2 × 48.4 kg × (velocity)^2
Step 3: Equate the potential energy and kinetic energy to find the velocity at the bottom.
Since the total mechanical energy is conserved, we can set the potential energy equal to the kinetic energy.
PE = KE
48.4 kg × 9.8 m/s^2 × 7.15 m = 1/2 × 48.4 kg × (velocity)^2
Simplifying the equation:
3359.84 J = 24.2 kg × (velocity)^2
Step 4: Solve for velocity.
Divide both sides of the equation by 24.2 kg:
velocity^2 = 3359.84 J / 24.2 kg
velocity^2 = 138.815703
velocity ≈ √138.815703
Therefore, the speed of the sled at the bottom of the hill is approximately 11.8 m/s.