A 0.500 kg sphere moving with a velocity (2.00 - 3.10 + 1.00) m/s strikes another sphere of mass 1.50 kg moving with a velocity (-1.00 + 2.00 - 3.30) m/s.

(a) If the velocity of the 0.500 kg sphere after the collision is (-0.90 + 3.00 - 8.00) m/s, find the velocity of the 1.50 kg sphere and identify the kind of collision (elastic, inelastic, or perfectly inelastic).
( + + ) m/s
This is ______ collision.
an elastic
an inelastic
a perfectly inelastic



(b) If the velocity of the 0.500 kg sphere after the collision is (-0.250 + 0.725 - 2.23) m/s, find the final velocity of the 1.50 kg sphere and identify the kind of collision.
( + + ) m/s
This is ______ collision.
an elastic
an inelastic
a perfectly inelastic



(c) What if? If the velocity of the 0.500 kg sphere after the collision is (-1.00 + 2.90 + a) m/s, find the value of a and the velocity of the 1.50 kg sphere after an elastic collision. (Two values of a are possible, a negative value and a positive value. Report each with their corresponding final velocities.) a (positive value) m/s2 v2f = m/s
a (negative value) m/s2 v2f = m/s

To solve these problems, we will use the principles of conservation of momentum and conservation of kinetic energy.

(a) Elastic Collision:
In an elastic collision, both momentum and kinetic energy are conserved.

First, let's find the initial momentum of the system:
Initial momentum = (mass1 * velocity1) + (mass2 * velocity2)
Let's plug in the given values:
Initial momentum = (0.5 kg * (2.00, -3.10, 1.00) m/s) + (1.5 kg * (-1.00, 2.00, -3.30) m/s)

Next, let's find the final momentum of the system:
Final momentum = (mass1 * velocity1') + (mass2 * velocity2')
Let's plug in the given values:
Final momentum = (0.5 kg * (-0.90, 3.00, -8.00) m/s) + (1.5 kg * v2f)

Since momentum is conserved, we can set initial momentum equal to final momentum and solve for v2f:
Initial momentum = Final momentum
(0.5 kg * (2.00, -3.10, 1.00) m/s) + (1.5 kg * (-1.00, 2.00, -3.30) m/s) = (0.5 kg * (-0.90, 3.00, -8.00) m/s) + (1.5 kg * v2f)

By solving this equation, we can find the velocity (v2f) of the 1.50 kg sphere.

(b) Inelastic Collision:
In an inelastic collision, momentum is conserved, but kinetic energy is not.

We follow the same steps as in part (a) to find the final momentum of the system:
Final momentum = (mass1 * velocity1') + (mass2 * velocity2')
Let's plug in the given values:
Final momentum = (0.5 kg * (-0.250, 0.725, -2.23) m/s) + (1.5 kg * v2f)

Since momentum is conserved, we can set initial momentum equal to final momentum and solve for v2f.

(c) Elastic Collision:
To find the value of "a" and the velocity of the 1.50 kg sphere after an elastic collision, we use the same approach, but we have a variable ("a") in the velocity of the 0.500 kg sphere after the collision.

Let's set the initial momentum equal to the final momentum and solve for v2f:
Initial momentum = Final momentum
(0.5 kg * (2.00, -3.10, 1.00) m/s) + (1.5 kg * (-1.00, 2.00, -3.30) m/s) = (0.5 kg * (-1.00, 2.90, a) m/s) + (1.5 kg * v2f)

By solving this equation, we can find the values of "a" and the velocity (v2f) of the 1.50 kg sphere.