A 26.3 g marble sliding to the right at 21.0 cm/s overtakes and collides elastically with a
13.3 g marble moving in the same direction at 12.2 cm/s. After the collision, the 13.3 g marble moves to the right at 23.7 cm/s.
Find the velocity of the 26.3 g marble after the collision.
To solve this problem, we can apply the conservation of momentum and conservation of kinetic energy principles.
1. Conservation of Momentum:
The total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
where:
m1 and m2 are the masses of the marbles,
v1 and v2 are the initial velocities of the marbles,
v1' and v2' are the velocities of the marbles after the collision.
2. Conservation of Kinetic Energy:
The total kinetic energy before the collision is equal to the total kinetic energy after the collision, since it's an elastic collision. Mathematically, this can be expressed as:
(1/2 * m1 * (v1^2)) + (1/2 * m2 * (v2^2)) = (1/2 * m1 * (v1'^2)) + (1/2 * m2 * (v2'^2))
where:
m1 and m2 are the masses of the marbles,
v1 and v2 are the initial velocities of the marbles,
v1' and v2' are the velocities of the marbles after the collision.
Now, let's substitute the given values into these equations:
m1 = 26.3 g = 0.0263 kg
v1 = 21.0 cm/s = 0.21 m/s
m2 = 13.3 g = 0.0133 kg
v2 = 12.2 cm/s = 0.122 m/s
v2' = 23.7 cm/s = 0.237 m/s
Using the above values, we can solve for v1':
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
(0.0263 kg * 0.21 m/s) + (0.0133 kg * 0.122 m/s) = (0.0263 kg * v1') + (0.0133 kg * 0.237 m/s)
0.0055239 kg·m/s + 0.0016266 kg·m/s = (0.0263 kg * v1') + 0.0031441 kg·m/s
0.0071505 kg·m/s = 0.0263 kg * v1' + 0.0031441 kg·m/s
0.0071505 kg·m/s - 0.0031441 kg·m/s = 0.0263 kg * v1'
0.0040064 kg·m/s = 0.0263 kg * v1'
Dividing both sides by 0.0263 kg:
v1' = 0.154 m/s
Therefore, the velocity of the 26.3 g marble after the collision is 0.154 m/s to the right.