A 60 kg person and an 75 kg person each jump off opposites sides of a 40 kg raft with a speed of 3.0 m/s. If the raft was initially at rest, what is the final speed of the raft? Was this collision elastic? Explain.
To find the final speed of the raft after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object is calculated by multiplying its mass by its velocity. In this case, the initial momentum of the system (person + raft) is zero since the raft was initially at rest.
Before we continue, let's assign some variables:
- The mass of person 1 = m1 = 60 kg
- The mass of person 2 = m2 = 75 kg
- The mass of the raft = m_r = 40 kg
- The initial velocity of person 1 = v1 = -3.0 m/s (opposite direction to person 2)
- The initial velocity of person 2 = v2 = 3.0 m/s (opposite direction to person 1)
- The final velocity of the raft = V_r
Using the conservation of momentum, we can write the equation:
(m1 * v1) + (m2 * v2) + (m_r * 0) = 0
Simplifying the equation gives us:
(m1 * v1) + (m2 * v2) = 0
Let's plug in the values:
(60 kg * -3.0 m/s) + (75 kg * 3.0 m/s) = 0
After calculating, we find that the total momentum before the collision is -180 kg·m/s + 225 kg·m/s = 45 kg·m/s.
Since the raft and the two people are the only objects involved in the collision, the total momentum after the collision is also 45 kg·m/s.
Let's represent the final velocity of the raft as V_r. Since the initial velocity of the raft was 0 m/s, the final momentum of the raft (m_r * V_r) will also be 45 kg·m/s. Therefore, we can replace the mass and velocity of the raft in the equation:
(40 kg * V_r) = 45 kg·m/s
Solving for V_r:
V_r = 45 kg·m/s / 40 kg
V_r = 1.125 m/s
So, the final speed of the raft after the collision is 1.125 m/s.
Now, to determine if the collision was elastic, we need to analyze whether kinetic energy was conserved. In an elastic collision, both momentum and kinetic energy are conserved.
The kinetic energy before the collision can be calculated by using the formula:
KE = (1/2) * mass * velocity^2
For person 1:
KE1 = (1/2) * 60 kg * (3.0 m/s)^2 = 270 J
For person 2:
KE2 = (1/2) * 75 kg * (3.0 m/s)^2 = 337.5 J
The total kinetic energy before the collision is:
KE_total = KE1 + KE2 = 270 J + 337.5 J = 607.5 J
After the collision, the total kinetic energy will be the sum of the kinetic energy of the raft and the two people.
For the raft:
KE_r = (1/2) * 40 kg * (1.125 m/s)^2 = 25.3125 J
For person 1 (after the collision):
KE1' = (1/2) * 60 kg * (-1.125 m/s)^2 = 37.5 J
For person 2 (after the collision):
KE2' = (1/2) * 75 kg * (1.125 m/s)^2 = 47.65625 J
The total kinetic energy after the collision is:
KE_total' = KE_r + KE1' + KE2' = 25.3125 J + 37.5 J + 47.65625 J = 110.46875 J
Since the total kinetic energy before the collision (607.5 J) is not equal to the total kinetic energy after the collision (110.46875 J), we can conclude that the collision was not elastic. Some of the kinetic energy was lost during the collision.
In summary, the final speed of the raft after the collision is 1.125 m/s, and the collision was not elastic as some kinetic energy was lost.