A 2 kg cart rolling at 3 m/s catches up to a 1 kg cart rolling the same way at 1.5 m/s, and the couplers latch. The pair rolls on at one shared speed. The conservation equation hands you that speed in two steps.
Take rightward as positive throughout the scene. For a coupling pair, the set-up is m_1 v_1 + m_2 v_2 = (m_1 + m_2) v', with one unknown: the shared final velocity v'.
Run the state table for the catching-up carts: the 2 kg cart at +3 m/s and the 1 kg cart at +1.5 m/s before; one combined 3 kg object at v' after.
Total the before side: the 2 kg cart carries 2 × 3 = 6 kg·m/s, the 1 kg cart carries 1 × 1.5 = 1.5 kg·m/s, and 6 + 1.5 = 7.5 kg·m/s.
The after side is the combined mass times the shared velocity: 3 v'.
Set the totals equal and divide: v' = 7.5 ÷ 3 = 2.5, so the coupled pair rolls on at 2.5 m/s rightward.
Head-on cases change nothing but the signs: a leftward mover enters with a negative velocity, and the before total can shrink toward zero.
A positive v' means the pair moves in the positive direction; the division hands the sign over automatically.
Take rightward as positive. A 3 kg cart moving right at 4 m/s collides head-on with a 1 kg cart moving left at 8 m/s, and the carts couple. Calculate the pair's velocity after the collision.
Write down the values in the question:
m_1 = 3 kg, v_1 = +4 m/s
m_2 = 1 kg, v_2 = −8 m/s
v' = ?
Write down the equation:
m_1 v_1 + m_2 v_2 = (m_1 + m_2) v'
Substitute in the values:
3 × 4 + 1 × (−8) = 4 v'
12 − 8 = 4 v'
Simplify:
v' = 4 ÷ 4 = 1
v' = +1 m/s
The coupled pair moves at 1 m/s in the positive (rightward) direction.
Take rightward as positive. A 1200 kg car rolling right at 2.5 m/s hits a parked 800 kg car, and the bumpers lock. Calculate the pair's velocity just after the impact.
Write down the values in the question:
m_1 = 1200 kg, v_1 = +2.5 m/s
m_2 = 800 kg, v_2 = 0 m/s
v' = ?
Write down the equation:
m_1 v_1 + m_2 v_2 = (m_1 + m_2) v'
Substitute in the values:
1200 × 2.5 + 0 = 2000 v'
3000 = 2000 v'
Simplify:
v' = 3000 ÷ 2000 = 1.5
v' = +1.5 m/s
The locked pair moves at 1.5 m/s in the positive (rightward) direction.
Take rightward as positive. A 4 kg cart moving right at 3 m/s catches a 2 kg cart moving right at 1.2 m/s, and they couple. Enter the pair's velocity after the collision in m/s.
Take rightward as positive. A 2 kg cart moving right at 4.5 m/s hits a parked 3 kg cart, and the two couple. Enter the pair's velocity after the collision in m/s.
Take rightward as positive. A 6 kg cart moving right at 2 m/s collides head-on with a 2 kg cart moving left at 4 m/s, and the carts couple. Enter the pair's velocity after the collision in m/s (include the sign).
Take rightward as positive. A 5 kg cart moving right at 3 m/s collides head-on with a 3 kg cart moving left at 3 m/s, and the carts couple. Enter the pair's velocity after the collision in m/s (include the sign).
Take rightward as positive. A 1500 kg van rolling right at 4.4 m/s hits a parked 500 kg car, and the bumpers lock. Enter the pair's velocity just after the impact in m/s.
Two carts couple in a collision. Which mass divides the before total of momentum to give the shared final velocity?