Two identical particles, each of mass 1400 kg, are coasting in free space along the same path. At one instant their separation is 30.0 m and each has precisely the same velocity of 820 m/s. What are their velocities when they are 4.00 m apart?
There is something contradicotry or incomplete about your question. If they have the same velocity and same psth at one time, in order for the separation to change, they must be accelerating at different rates. Yet you say that both are "cossting in free space"
This makes no sense to me
I know!
but that's the exact homework question I have.
I think it means that at that instant (when they're 30 ft apart) they have the same velocity. But mass 2 is gaining on mass 1. But I could be mistaken.
To solve this problem, we can use the principle of conservation of momentum. The total momentum of a system remains constant if no external forces are acting on it.
Initially, the particles have the same velocity and are traveling along the same path. The momenta of the two particles are equal in magnitude and opposite in direction, so the total momentum of the system is zero.
At one instant, their separation is 30.0 m. We are given that each particle has a mass of 1400 kg and a velocity of 820 m/s. We can calculate the total initial momentum of the system:
Initial momentum = mass × velocity
Initial momentum = (mass of particle 1 × velocity of particle 1) + (mass of particle 2 × velocity of particle 2)
Since the particles have the same mass and velocity:
Initial momentum = (1400 kg × 820 m/s) + (1400 kg × 820 m/s)
Initial momentum = 2 × (1400 kg × 820 m/s)
Initial momentum = 2 × (1400 kg × 820 m/s)
Initial momentum = 2 × 1148000 kg·m/s
Initial momentum = 2296000 kg·m/s
Now, we need to find the final velocities when the particles are 4.00 m apart. Since the particles are identical, their final velocities will have the same magnitude and be opposite in direction.
Let's assume one particle has a final velocity of v and the other particle has a final velocity of -v. The final separation between the particles is 4.00 m.
Using the principle of conservation of momentum:
Initial momentum = Final momentum
(Initial mass × Initial velocity) + (Initial mass × Initial velocity) = (Final mass × Final velocity) + (Final mass × Final velocity)
Since the particles have the same mass:
Initial momentum = (Final mass × Final velocity) + (Final mass × -Final velocity)
2296000 kg·m/s = (2 × Final mass) × (Final velocity - Final velocity)
2296000 kg·m/s = 0
From this equation, we see that the final total momentum of the system is zero. Therefore, the final velocities of the particles when they are 4.00 m apart are both 0 m/s.
So, their velocities when they are 4.00 m apart are both 0 m/s.