An 67.6 kg object moving to the right at
38.7 cm/s overtakes and collides elastically with a second 48.9 kg object moving in the same direction at 19.2 cm/s.
Find the velocity of the second object after the collision.
Answer in units of cm/s
To find the velocity of the second object after the collision, we need to apply the principles of conservation of momentum and conservation of kinetic energy.
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be written as:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
where:
m1 and m2 are the masses of the two objects,
v1 and v2 are the initial velocities of the two objects,
v1' and v2' are the final velocities of the two objects after the collision.
In this case, the first object is m1 = 67.6 kg, moving at v1 = 38.7 cm/s, and the second object is m2 = 48.9 kg, moving at v2 = 19.2 cm/s.
To solve for v2', we can rearrange the equation as:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
(m1 * v1) + (m2 * v2) - (m1 * v1') = (m2 * v2')
(m1 * v1) + (m2 * v2) - (m1 * v1') = m2 * v2'
(m1 * v1) + (m2 * v2) - m1 * v1' = m2 * v2'
v2' = [(m1 * v1) + (m2 * v2) - m1 * v1'] / m2
Now, we can substitute the given values into the equation and solve for v2':
v2' = [(67.6 kg * 38.7 cm/s) + (48.9 kg * 19.2 cm/s) - (67.6 kg * v1')] / 48.9 kg
Now, we need the value of v1'. Since the collision is elastic, we know that kinetic energy is conserved.
The total kinetic energy before the collision is:
KE1 = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
The total kinetic energy after the collision is:
KE1' = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2
Since the collision is elastic, KE1 = KE1', and we can equate the two equations:
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2
Simplifying the equation above, we get:
m1 * v1^2 + m2 * v2^2 = m1 * v1'^2 + m2 * v2'^2
Substituting the given values, we have:
(67.6 kg * (38.7 cm/s)^2) + (48.9 kg * (19.2 cm/s)^2) = (67.6 kg * v1'^2) + (48.9 kg * v2'^2)
Now we have two equations:
(67.6 kg * 38.7 cm/s) + (48.9 kg * 19.2 cm/s) - (67.6 kg * v1') = 48.9 kg * v2'
(67.6 kg * (38.7 cm/s)^2) + (48.9 kg * (19.2 cm/s)^2) = (67.6 kg * v1'^2) + (48.9 kg * v2'^2)
By solving these two equations together, we can find v2', which will be the velocity of the second object after the collision.