Question

Two carts, one of mass 2m and one of mass m, approach each other with the same speed, v. When the carts collide, they hook together. Assume that positive momentum is to the right.

A diagram of two rolling carts on a flat horizontal surface rolling toward each other at a velocity v. The cart on the left is labeled m and the cart on the right is labeled 2 m.
Which graph best represents the momentum of both carts over time, before and after the collision?

A graph with momentum on the y axis measured in kilogram meters per second and time on the x axis measured in seconds. There are no values on the axes. There are two curves on the graph. The first starts above the x axis and is horizontal before curving downward to the x axis labeled 2 m cart. The second curve starts below the axis, the same distance below as the first one was above, and is horizontal before curving upward toward the x axis at the same time and labeled m cart. Where they meet is labeled Collision. Both curves meet at the x axis at the same time and are both on the x axis for a time period labeled Combined carts.
A graph with momentum on the y axis measured in kilogram meters per second and time on the x axis measured in seconds. There are no values on the axes. There are two curves on the graph. The first starts above the x axis and is horizontal before curving downward to the x axis labeled 2 m cart. The second curve starts below the axis, the same distance below as the first one was above, and is horizontal before curving upward toward the x axis at the same time and labeled m cart. Where they meet is labeled Collision. Both curves meet at the x axis at the same time and are both on the x axis for a time period labeled Combined carts.
A graph with momentum on the y axis measured in kilogram meters per second and time on the x axis measured in seconds. There are no values on the axes. There are two curves on the graph the first starts above the x axis and is horizontal before curving downward toward the x axis labeled 2 m cart. The second curve starts below the axis, a smaller distance below as the first one was above, and is horizontal before curving upward toward and crosses the x axis and is labeled m cart. Where they meet is labeled Collision. Both curves meet above the axis at the same time and both go horizontal above the x axis for a time period labeled Combined carts.
A graph with momentum on the y axis measured in kilogram meters per second and time on the x axis measured in seconds. There are no values on the axes. There are two curves on the graph the first starts above the x axis and is horizontal before curving downward toward the x axis labeled 2 m cart. The second curve starts below the axis, a smaller distance below as the first one was above, and is horizontal before curving upward toward and crosses the x axis and is labeled m cart. Where they meet is labeled Collision. Both curves meet above the axis at the same time and both go horizontal above the x axis for a time period labeled Combined carts.

Answer 1

None of the four graphs is correct.

Reason and correct shape:
- Take right as positive. Initial momenta:
- left (mass m moving right): p_left = +m v
- right (mass 2m moving left): p_right = −2m v
- total: p_total = +m v + (−2m v) = −m v (constant at all times)
- After they stick the combined mass is 3m, so common velocity = (−m v)/(3m) = −v/3. Individual momenta after collision:
- p_left(after) = m(−v/3) = −(1/3) m v
- p_right(after) = 2m(−v/3) = −(2/3) m v
Both are negative (left has crossed from positive to negative; right remains negative but with reduced magnitude).
- Therefore the correct graph:
- Before collision: two horizontal lines at +m v (left) and −2m v (right).
- During collision: the left-cart line slopes downward, crossing the time axis (goes from +m v to −(1/3) m v); the right-cart line slopes upward but remains below the axis (from −2m v to −(2/3) m v).
- After collision: two horizontal lines below the axis at −(1/3) m v and −(2/3) m v. The total (combined) momentum is a single horizontal line at −m v the whole time.

None of the provided options shows this pattern (left-cart crossing from positive to negative while the right-cart stays negative and both end up below the axis with the correct magnitudes).