A lab cart carries an unlabeled load, and the balance is across the room. A known cart rolls into it, the sensors read the velocities, and the collision itself weighs the load.
In the conservation equation, the masses and the velocities share one bookkeeping. When every velocity is measured and one mass is missing, the same equation solves for the mass.
Here is the scene, taking rightward as positive: a 2 kg cart moving right at 6 m/s hits the unknown cart, parked. They couple and roll off together at 1.5 m/s.
The before total is 2 × 6 = 12, so the system carries 12 kg·m/s.
After coupling, the combined mass carries that total at 1.5 m/s. So the combined mass is 12 ÷ 1.5 = 8 kg.
The combined mass is the two masses added. Subtract the known 2 kg: the unknown cart's mass is 8 − 2 = 6 kg.
The same bookkeeping works on a separation. Two carts push apart from rest: the known cart's momentum is measured, the unknown cart's velocity is measured, and the unknown mass is the momentum size divided by that velocity.
A mass must come out positive. A negative mass means a sign went in wrong, so check the velocities against the convention.
Take rightward as positive. A 3 kg cart and a cart of unknown mass push apart from rest. The 3 kg cart recoils at 2 m/s leftward, and the unknown cart moves at 1.5 m/s rightward. Calculate the unknown mass.
Write down the values in the question:
m_1 = 3 kg, v_1' = −2 m/s
m_2 = ?, v_2' = +1.5 m/s
Total momentum before = 0 kg·m/s (both at rest)
Write down the equation:
0 = m_1 v_1' + m_2 v_2'
Substitute in the values:
0 = 3 × (−2) + m_2 × 1.5
0 = −6 + 1.5 m_2
Simplify:
m_2 = 6 ÷ 1.5 = 4
m_2 = 4 kg
The unknown cart's mass is 4 kg.
Take rightward as positive. A 1 kg cart moving right at 5 m/s hits a parked cart of unknown mass, and the two couple, rolling off at 0.5 m/s. Calculate the unknown mass.
Write down the values in the question:
m_1 = 1 kg, v_1 = +5 m/s
m_2 = ?, v_2 = 0 m/s, shared v' = +0.5 m/s
Write down the equation:
m_1 v_1 + m_2 v_2 = (m_1 + m_2) v'
Substitute in the values:
1 × 5 + 0 = (1 + m_2) × 0.5
5 = 0.5 + 0.5 m_2
Simplify:
0.5 m_2 = 5 − 0.5 = 4.5
m_2 = 4.5 ÷ 0.5 = 9
m_2 = 9 kg
The unknown cart's mass is 9 kg.
Take rightward as positive. A 2 kg cart moving right at 4.5 m/s hits a parked cart of unknown mass, and the two couple, rolling off at 1.2 m/s. Enter the unknown mass in kg.
Take rightward as positive. A 4 kg cart and a cart of unknown mass push apart from rest. The 4 kg cart recoils at 3.5 m/s leftward, and the unknown cart moves at 2 m/s rightward. Enter the unknown mass in kg.
Take rightward as positive. A 4 kg cart moving right at 3 m/s hits a parked cart of unknown mass, and they bounce apart. Afterward the 4 kg cart moves at +1 m/s and the unknown cart at +3.2 m/s. Enter the unknown mass in kg.
Take rightward as positive. A 1500 kg van rolling right at 3 m/s hits a parked car of unknown mass, and the bumpers lock. The pair slides off at 1.8 m/s. Enter the car's mass in kg.
Take rightward as positive. A 0.6 kg cart and a cart of unknown mass push apart from rest. The 0.6 kg cart recoils at 5 m/s leftward, and the unknown cart moves at 4 m/s rightward. Enter the unknown mass in kg.
A known cart of mass m_1 moving at v_1 hits a parked cart, and the pair couples and moves at v'. Which expression gives the parked cart's mass?