hange the mass of the red ball to match that of the green ball.
a. Which ball has greater momentum now?
b. How has the total momentum changed?
c. Predict what will happen to the motion of the balls after they collide.

Question

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$$ \textbf{a. Which ball has greater momentum now?} $$

Momentum is given by 
$$
p = m \, v.
$$
If we now set the mass of the red ball equal to the mass of the green ball (let’s call this common mass $$ m $$), then each ball’s momentum becomes

$$
p_{\text{red}} = m \, v_{\text{red}} \quad \text{and} \quad p_{\text{green}} = m \, v_{\text{green}}.
$$

Thus, with equal masses the magnitude of momentum depends solely on the speed. If, for example, the green ball is moving faster (i.e. $$ v_{\text{green}} > v_{\text{red}} $$), then

$$
p_{\text{green}} > p_{\text{red}}.
$$

So the ball with the higher speed has the greater momentum.

---

$$ \textbf{b. How has the total momentum changed?} $$

Assume that before this change the red ball had a different mass, say $$ m_{\text{red,orig}} $$, and its momentum was

$$
p_{\text{red,orig}} = m_{\text{red,orig}} \, v_{\text{red}},
$$
while the green ball had momentum

$$
p_{\text{green}} = m \, v_{\text{green}}.
$$

After changing the red ball’s mass to $$ m $$ (equal to the green ball’s mass), its momentum becomes

$$
p_{\text{red,new}} = m \, v_{\text{red}}.
$$

If the original mass $$ m_{\text{red,orig}} $$ was larger than $$ m $$, then

$$
p_{\text{red,new}} < p_{\text{red,orig}},
$$

and therefore the total momentum of the system,

$$
p_{\text{total}} = p_{\text{red}} + p_{\text{green}},
$$

will have decreased by the reduction in the red ball’s momentum. (If no external forces act and the speeds were maintained during the change, then the decrease in mass directly decreases momentum.)

---

$$ \textbf{c. Predict what will happen to the motion of the balls after they collide.} $$

In collisions, momentum is conserved (if no external forces act). With both balls having the same mass, the outcome of the collision will depend largely on the type of collision (elastic or inelastic) and their initial speeds. In many cases (for example, in an elastic collision with one ball moving faster than the other):

- The faster (green) ball will transfer some of its momentum to the slower (red) ball.
- If the collision is head‐on and elastic, the moving ball may slow down considerably or even come nearly to rest, while the red ball gains speed.
- The precise outcome will obey conservation of both momentum and energy (in an elastic collision) and might resemble collisions of billiard balls where equal-mass objects exchange speeds.

Thus, after the collision you can expect an exchange of speeds and momentum between the balls, leading to a change in their motions consistent with the principles of momentum conservation.
a. The ball with the higher speed has greater momentum. b. The total momentum has decreased because the red ball now has less momentum than before. c. After colliding, the green ball will slow down, and the red ball will speed up, exchanging their momenta.

hange the mass of the red ball to match that of the green ball.
a. Which ball has greater momentum now?
b. How has the total momentum changed?
c. Predict what will happen to the motion of the balls after they collide.

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hange the mass of the red ball to match that of the green ball. a. Which ball has greater momentum now? b. How has the total momentum changed? c. Predict what will happen to the motion of the balls after they collide.

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hange the mass of the red ball to match that of the green ball.
a. Which ball has greater momentum now?
b. How has the total momentum changed?
c. Predict what will happen to the motion of the balls after they collide.