gayle runs at a speed of 4.00 m/s and dives on a sled, initially at rest on the top of a frictionless, snow-covered hill. after she has descended a vertical distance at a 5.00 m, her brother,who is initially at rest, hops on her back, and they continue down the hill together, what is their speed at the buttom of the hill if the total vertical drop is 15.0 m? gayle's mass is 50.0 kg, the sled has a mass of 5.00 kg, and her brother 's mass is 30.0 kg.
To determine the speed of Gayle and her brother at the bottom of the hill, you can use the principle of conservation of energy. The initial potential energy of the system (Gayle, her brother, and the sled) when they are at the top of the hill is equal to the final kinetic energy of the system at the bottom of the hill.
At the top of the hill, the system only has potential energy, which is given by:
Potential Energy = Mass * Gravity * Height
Here, the mass is the combined mass of Gayle, her brother, and the sled (50.0 kg + 30.0 kg + 5.00 kg) = 85.0 kg.
The acceleration due to gravity is approximately 9.8 m/s².
The height is 5.00 m (distance Gayle travels before her brother hops on).
Potential Energy = 85.0 kg * 9.8 m/s² * 5.00 m = 4165 J
Now, at the bottom of the hill, the system will have both kinetic energy and potential energy. The potential energy becomes zero since they are at the lowest point, while the final kinetic energy is given by:
Kinetic Energy = (1/2) * Mass * Velocity²
Here, the mass is still the combined mass of Gayle, her brother, and the sled (85.0 kg).
We need to find the final velocity, so we can rearrange the equation to solve for it:
Velocity = √( (2 * Kinetic Energy) / Mass)
Since the potential energy becomes zero, the total energy (potential energy + kinetic energy) at the bottom is equal to the initial potential energy:
Kinetic Energy + 0 = 4165 J
Simplifying, we have:
Kinetic Energy = 4165 J
Velocity = √( (2 * 4165 J) / 85.0 kg)
Calculating this expression gives us the final velocity.