a 50 kg person running to the east at 8.5 m/s jumps into a 100 kg initially at rest. A)find the velocity of the wagon and person together as they move away.B) If they coast 30 m before coming to a stop, find the average force of the friction that slows them down.
To find the answers for both parts of the question, we can apply the principle of conservation of momentum and the laws of motion.
A) To find the velocity of the wagon and person together as they move away, we can make use of the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision, provided there are no external forces acting on the system.
The momentum of an object is given by the product of its mass and velocity. We can calculate the total momentum before and after the collision as follows:
Momentum before collision = (mass of person) × (velocity of person) + (mass of wagon) × (velocity of wagon before collision)
Momentum after collision = (total mass of person and wagon) × (velocity of person and wagon together after collision)
Given:
Mass of person (m1) = 50 kg
Velocity of person (v1) = 8.5 m/s
Mass of wagon (m2) = 100 kg
Velocity of wagon before collision (v2) = 0 m/s (since it is initially at rest)
Velocity of person and wagon together after collision (v3) = ?
Using the principle of conservation of momentum:
(m1 × v1) + (m2 × v2) = (m1 + m2) × v3
Substituting the known values:
(50 kg × 8.5 m/s) + (100 kg × 0 m/s) = (50 kg + 100 kg) × v3
425 kg·m/s = 150 kg × v3
We can now solve for v3:
v3 = (425 kg·m/s) / (150 kg)
v3 ≈ 2.83 m/s
Therefore, the velocity of the wagon and person together as they move away is approximately 2.83 m/s.
B) To find the average force of friction that slows them down over a distance of 30 m, we can use the work-energy principle. By applying this principle, we can relate the work done by the frictional force to the initial and final kinetic energies of the system.
The work done by the frictional force is equal to the change in kinetic energy:
Work (W) = Change in Kinetic Energy (ΔKE)
The initial kinetic energy of the system can be calculated as:
Initial Kinetic Energy (KE1) = (1/2) × Total Mass × (Velocity of person and wagon together after collision)^2
Substituting the known values:
KE1 = (1/2) × (150 kg) × (2.83 m/s)^2
The final kinetic energy of the system is zero because they come to a stop. Therefore, KE2 = 0.
The work done by friction can be calculated as:
W = KE2 - KE1
Substituting the values:
W = 0 - [(1/2) × (150 kg) × (2.83 m/s)^2]
W = - [(1/2) × (150 kg) × (2.83 m/s)^2]
The negative sign indicates that work is done against the direction of motion.
Finally, we can use the definition of work to calculate the force of friction (F):
W = F × Distance (d)
- [(1/2) × (150 kg) × (2.83 m/s)^2] = F × 30 m
Solving for F:
F = - [(1/2) × (150 kg) × (2.83 m/s)^2] / 30 m
Therefore, the average force of friction that slows them down over a distance of 30 m is the calculated value of F.