A person of mass 84.5 kg is initially at rest on the edge of a large stationary platform of mass 150 kg, supported by frictionless wheels on a horizontal surface. The person jumps off the platform, traveling a horizontal distance of 1.00 m while falling a vertical distance of 0.500 m to the ground. What is the final speed of the platform?
To find the final speed of the platform, we need to apply the principle of conservation of momentum.
Conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it. In this scenario, the person and the platform can be considered as an isolated system.
The initial momentum of the system is zero since both the person and the platform are at rest. Therefore, the total momentum after the person jumps off should also be zero.
Let's denote the velocity of the platform as Vp and the velocity of the person as Vp'. The mass of the platform is 150 kg, and the mass of the person is 84.5 kg.
The momentum of an object can be calculated by multiplying its mass by its velocity:
Initial momentum of the system = (mass of the platform + mass of the person) × 0 (as both are initially at rest)
Final momentum of the system = mass of the platform × Vp + mass of the person × Vp'
Since the final momentum is required to be zero, we can write:
mass of the platform × Vp + mass of the person × Vp' = 0
Substituting the known values:
150 kg × Vp + 84.5 kg × Vp' = 0
Now, we need to find the relationship between Vp and Vp' by considering the conservation of energy. When the person jumps off the platform, the gravitational potential energy is converted into the kinetic energy of the person and the platform.
The gravitational potential energy can be calculated as:
Gravitational potential energy = mass of the person × gravity × height
In this case, the mass of the person is 84.5 kg, the height is 0.500 m, and the acceleration due to gravity is approximately 9.8 m/s^2.
Gravitational potential energy = 84.5 kg × 9.8 m/s^2 × 0.500 m
Now, let's equate the gravitational potential energy to the kinetic energy:
Gravitational potential energy = Kinetic energy
84.5 kg × 9.8 m/s^2 × 0.500 m = (1/2)(mass of the platform)(Vp)^2 + (1/2)(mass of the person)(Vp')^2
Substituting the known values:
409.21 J = (1/2)(150 kg)(Vp)^2 + (1/2)(84.5 kg)(Vp')^2
We also know that the distance traveled by the person horizontally (1.00 m) is equal to the distance traveled by the platform:
Distance = velocity × time
1.00 m = Vp × time
Since the person and the platform jump at the same time, their time of flight is equal. Therefore, we can write:
Distance = Vp' × time
Substituting the value of Vp' in the above equation:
1.00 m = (Vp' × time)
Now, we have two equations that relate Vp and Vp' to time:
150 kg × Vp + 84.5 kg × Vp' = 0 ------- (Equation 1)
1.00 m = (Vp' × time) ----------- (Equation 2)
Solving these two equations simultaneously will give the final value of Vp.
Please note that the time of flight is not given in the problem statement. Therefore, we would require additional information or assumptions to find the final speed of the platform accurately.