In an elastic 1 -dimensional collision, a mass m1=3 kg with velocity v1 = -4 m/s collides with a mass m2 = 5kg at rest. What are the velocities v1' and v2' after the collision? I would like to your method of solving this problem. Thanks.
conserve momentum and KE before and after collision
(3)(-4) + (5)(0) = 3(v1)+5(v2)
(1/2)(3)(16) + (1/2)(5)(0) = (1/2)(3)(v1^2) + (1/2)(5)(v2^2)
v1 = 1
v2 = -3
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of momentum:
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. Mathematically,
m1 * v1 + m2 * v2 = m1 * v1' + m2 * v2'
where m1 and m2 are the masses of the objects, v1 and v2 are their initial velocities, and v1' and v2' are their final velocities.
2. Conservation of kinetic energy:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Mathematically,
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2
Applying these principles, we can solve for v1' and v2'.
Given:
m1 = 3 kg
v1 = -4 m/s
m2 = 5 kg
v2 = 0 m/s (at rest)
Step 1: Conservation of momentum:
3 kg * (-4 m/s) + 5 kg * 0 m/s = 3 kg * v1' + 5 kg * v2'
Simplifying the equation:
-12 kg m/s = 3 kg * v1' + 0 kg m/s
-12 kg m/s = 3 kg * v1'
Dividing both sides by 3 kg:
-4 m/s = v1'
So, the final velocity of mass m1 (v1') is -4 m/s.
Step 2: Conservation of kinetic energy:
(1/2) * 3 kg * (-4 m/s)^2 + (1/2) * 5 kg * 0 m/s^2 = (1/2) * 3 kg * (-4 m/s)^2 + (1/2) * 5 kg * v2'^2
Simplifying the equation:
24 J = 24 J + (1/2) * 5 kg * v2'^2
24 J = 24 J + 5/2 kg * v2'^2
Subtracting 24 J from both sides:
0 J = 5/2 kg * v2'^2
Since the initial kinetic energy is not equal to zero, the kinetic energy after the collision must also be non-zero. Thus, v2' must be non-zero.
Dividing both sides by (5/2) kg:
0 J / (5/2) kg = v2'^2
0 m^2/s^2 = v2'^2
Taking the square root of both sides:
v2' = 0 m/s
So, the final velocity of mass m2 (v2') is 0 m/s.
Therefore, after the collision, mass m1 has a final velocity of -4 m/s and mass m2 has a final velocity of 0 m/s.