A lab experiment hands you a table: masses and velocities measured before and after two carts collide. Did the collision conserve the system's momentum? The verdict is not printed anywhere in the table. It comes out of two totals.
The routine has three steps: total the momenta before, total the momenta after, and compare. Signs come first, as always: declare the positive direction before touching a number.
Here is a measured run. Take rightward as positive. Before the collision, cart A (2.0 kg) moves at +3.0 m/s and cart B (2.0 kg) at +0.5 m/s. After it, A moves at +1.5 m/s and B at +2.0 m/s.
Total before: p_A = +6.0 kg·m/s and p_B = +1.0 kg·m/s, so the total is +7.0 kg·m/s.
Total after: p_A = +3.0 kg·m/s and p_B = +4.0 kg·m/s, so the total is +7.0 kg·m/s.
The totals match. The collision conserved the system's momentum.
The bar charts below say the same thing in one look: the carts trade momentum, and the total bar does not move.
Not every run passes. If an external force acts on the system during the run, the after-total comes out different, and the verdict is not conserved.
Take rightward as positive. Before a collision, a 4.0 kg sled moves at +2.5 m/s and a 0.5 kg sled at −4.0 m/s. After it, the 4.0 kg sled moves at +1.4 m/s and the 0.5 kg sled at +1.2 m/s. Was the system's momentum conserved?
Write down the values in the question:
Take rightward as positive.
before: m_A = 4.0 kg at +2.5 m/s; m_B = 0.5 kg at −4.0 m/s
after: m_A = 4.0 kg at +1.4 m/s; m_B = 0.5 kg at +1.2 m/s
Total the momenta before:
p_A = +10 kg·m/s, p_B = −2.0 kg·m/s
total before = 10 + (−2.0)
total before = +8.0 kg·m/s
Total the momenta after:
p_A = +5.6 kg·m/s, p_B = +0.6 kg·m/s
total after = 5.6 + 0.6
total after = +6.2 kg·m/s
Compare:
+8.0 kg·m/s before, +6.2 kg·m/s after: the totals differ
The totals differ, so the collision did not conserve the system's momentum. An external force acted during the run.
Take rightward as positive. Two wagons are measured before they collide: a 1.8 kg wagon moves at +4.0 m/s and a 1.4 kg wagon at −2.5 m/s. Enter the system's total momentum before the collision, in kg·m/s.
The same run, measured after the collision: the 1.8 kg wagon moves at +1.5 m/s and the 1.4 kg wagon at +0.5 m/s. Enter the system's total momentum after the collision, in kg·m/s.
The same run once more: compare your two totals, before and after the collision. Was the system's momentum conserved?
Take eastward as positive. Before a collision, a 2.4 kg cart moves at +2.5 m/s and a 1.6 kg cart at −2.5 m/s. After it, both carts move at +0.5 m/s. What do the data show about the system's total momentum through the collision?
Take rightward as positive. A 3.0 kg box slides at +2.0 m/s into a 2.0 kg box moving at +0.5 m/s on a rough floor. After the collision, the boxes move together at +1.1 m/s. What do the data show about the system's total momentum through the collision?
Take rightward as positive. Before a collision, a 2.5 kg cart moves at +2.0 m/s and a 0.5 kg cart at +2.0 m/s. After it, the 2.5 kg cart moves at +1.2 m/s and the 0.5 kg cart at +3.0 m/s. Calculate the change in the system's total momentum: enter the total after minus the total before, in kg·m/s.