Two objects collide head-on (see figure below). The first object is moving with an initial speed of
v1i = 8.04 m/s
and the second object is moving with an initial speed of
v2i = 10.00 m/s.
Assuming the collision is elastic,
m1 = 5.24 kg,
and
m2 = 6.21 kg,
determine the final velocity of each object.
To determine the final velocity of each object in an elastic collision, we can use the equation of conservation of momentum and the equation of conservation of kinetic energy.
1. Conservation of momentum:
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The formula for momentum is:
momentum = mass x velocity
For object 1:
momentum1_initial = m1 * v1i
For object 2:
momentum2_initial = m2 * v2i
After the collision, the final momentum of each object can be represented as:
momentum1_final = m1 * v1f
momentum2_final = m2 * v2f
Since the collision is head-on, the initial momentum of object 1 is in the opposite direction to the initial momentum of object 2. Therefore, our equation becomes:
momentum1_initial + momentum2_initial = momentum1_final + momentum2_final
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.
The formula for kinetic energy is:
kinetic energy = (1/2) * mass * velocity^2
For object 1:
kinetic_energy1_initial = (1/2) * m1 * v1i^2
For object 2:
kinetic_energy2_initial = (1/2) * m2 * v2i^2
After the collision, the final kinetic energy of each object can be represented as:
kinetic_energy1_final = (1/2) * m1 * v1f^2
kinetic_energy2_final = (1/2) * m2 * v2f^2
Since the collision is elastic, the equation becomes:
kinetic_energy1_initial + kinetic_energy2_initial = kinetic_energy1_final + kinetic_energy2_final
Now, we can solve these two equations simultaneously to find the final velocities of both objects.
1. Substitute the known values into the equations:
momentum1_initial = m1 * v1i = 5.24 kg * 8.04 m/s
momentum2_initial = m2 * v2i = 6.21 kg * 10.00 m/s
kinetic_energy1_initial = (1/2) * m1 * v1i^2 = (1/2) * 5.24 kg * (8.04 m/s)^2
kinetic_energy2_initial = (1/2) * m2 * v2i^2 = (1/2) * 6.21 kg * (10.00 m/s)^2
2. Rearrange the equations:
momentum1_initial + momentum2_initial = momentum1_final + momentum2_final
kinetic_energy1_initial + kinetic_energy2_initial = kinetic_energy1_final + kinetic_energy2_final
3. Substitute the equations into the rearranged equations:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
(1/2) * m1 * v1i^2 + (1/2) * m2 * v2i^2 = (1/2) * m1 * v1f^2 + (1/2) * m2 * v2f^2
4. Solve for v1f and v2f:
Use the above equations to solve for v1f and v2f. It involves algebraic manipulation and solving the resulting equations simultaneously.
By solving these equations, you will find the final velocity of each object after the collision.