Two particles approach each other with equal and opposite speed v. The mass of one particle is m, and the mass of the other particle is nm, where n is just a unitless number. Snapshots of the system before, during, and after the elastic collision are shown above. After the collision the first particle moves in the exact opposite direction with speed 1.30v, and the speed of the second particle is unknown. What is the value of n?
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy during elastic collisions.
Let's denote the speed of the second particle after the collision as u.
Before the collision:
The first particle has a mass of m and is moving with velocity v.
The momentum of the first particle before the collision is given by p = m * v.
The second particle has a mass of nm and is moving with velocity -v.
The momentum of the second particle before the collision is given by p = (nm) * (-v) = -n * m * v.
Since the particles have equal and opposite speed, their momenta cancel each other out before the collision:
m * v - n * m * v = 0
(m - n * m) * v = 0
During the collision:
Since the collision is elastic, the total kinetic energy of the system is conserved. The initial kinetic energy is given by 0.5 * m * v^2 + 0.5 * (nm) * v^2 = 0.5 * m * v^2 + 0.5 * n * m * v^2.
After the collision:
The first particle moves in the exact opposite direction with a speed of 1.30v. Its final momentum is given by p = m * 1.30v = 1.30mv.
The second particle moves with a speed of u. Its final momentum is given by p = (nm) * u = nmu.
By conservation of momentum, the sum of the final momenta is equal to the initial momentum:
1.30mv + nmu = 0
To find the value of n, we can solve the two equations we obtained:
(m - n * m) * v = 0
1.30mv + nmu = 0
Since v cannot be zero (otherwise the particles would not be approaching each other), we can divide both equations by v to simplify the problem:
m - n * m = 0
1.30m + n * u = 0
Let's solve the first equation for n:
m - n * m = 0
n * m = m
n = 1
Therefore, the value of n is 1.