Two carts are about to collide on a low-friction track, and you want the velocities after the impact. The law says the totals match. To use that law on numbers, you need it written as an equation you can solve.
The law of conservation of momentum: the total momentum of a system stays constant when the net external force on the system is zero. This lesson turns the law into a working equation, one signed value at a time.
Here is the scene, taking rightward as positive: a 2 kg cart moves right at 3 m/s toward a 4 kg cart moving left at 1.5 m/s.
A table with one row per cart and a velocity column for each moment keeps the bookkeeping honest, every value signed. A table like this is called a state table: it holds each cart's state, before and after.
The signs come from the standing convention: taking rightward as positive, a leftward velocity carries a negative sign. The 4 kg cart's velocity goes in as −1.5 m/s.
The law says the total momentum before equals the total momentum after. In symbols, for two objects: m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'.
The primes mark the after side: v_1' is one symbol, the first cart's velocity after the interaction.
For the scene above, the substituted equation reads: 2 × 3 + 4 × (−1.5) = 2 v_1' + 4 v_2'.
One more form matters. When the objects stick together, the after side has one shared velocity v', so the equation becomes m_1 v_1 + m_2 v_2 = (m_1 + m_2) v'.
Take rightward as positive. A 3 kg cart moving right at 2 m/s hits a 2 kg cart moving left at 0.5 m/s, and the two couple. Write the conservation equation for the collision, with the values substituted.
Write down the values in the question:
m_1 = 3 kg, v_1 = +2 m/s
m_2 = 2 kg, v_2 = −0.5 m/s (leftward, so negative)
The carts couple, so the after side is (m_1 + m_2) v'.
Write down the equation:
m_1 v_1 + m_2 v_2 = (m_1 + m_2) v'
Substitute in the values:
3 × 2 + 2 × (−0.5) = 5 v'
The set-up equation is 3 × 2 + 2 × (−0.5) = 5 v'.
Take eastward as positive. A 1 kg puck moving east at 4 m/s hits a 3 kg puck moving west at 2 m/s, and the pucks bounce apart. Write the conservation equation with the before values substituted.
Write down the values in the question:
m_1 = 1 kg, v_1 = +4 m/s
m_2 = 3 kg, v_2 = −2 m/s (westward, so negative)
The pucks separate, so the after side keeps two velocities.
Write down the equation:
m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'
Substitute in the values:
1 × 4 + 3 × (−2) = 1 v_1' + 3 v_2'
The set-up equation is 1 × 4 + 3 × (−2) = 1 v_1' + 3 v_2'.
Take rightward as positive. A 3 kg cart moves right at 2 m/s and a 1 kg cart moves left at 4 m/s. They collide and bounce apart. Which is the correct set-up?
Take eastward as positive. A 5 kg cart moves west at 2 m/s. Enter the value, in m/s, that this cart's velocity takes in the conservation equation (include the sign).
Oriana writes, for two carts that couple on impact: m_1 v_1 + m_2 v_2 = (m_1 + m_2) v'. Is her set-up correct?
Take rightward as positive. A 2.5 kg cart moves right at 4 m/s and a 1.5 kg cart moves left at 2 m/s. They collide and bounce apart. Which is the correct set-up?
Take rightward as positive. A 6 kg cart moves left at 2.5 m/s. Enter this cart's momentum, in kg·m/s, as it appears in the conservation equation (include the sign).
In m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2', which quantity does the left side give?