i am having trouble with this ap physics

Question

An archer tests various arrowheads by shooting arrows at a pumpkin that is suspended from a tree branch by a rope, as shown to the right. When struck head-on by the arrow, the pumpkin swings upward on the rope. The maximum angle θ that the rope makes with the vertical is different for each arrowhead that the archer tests. Each arrow, including its arrowhead, has the same mass m and is shot with the same velocity v₀ toward the right.

The arrowheads are made of different materials, however, and each behaves differently when it strikes the pumpkin, as described below.

1. Embedded arrow: Strikes the pumpkin and remains embedded, while the pumpkin swings to angle θ_emb.
2. Pass arrow: Passes all the way through the pumpkin and continues traveling away from the archer, while the pumpkin swings to angle θ_pass.
3. Bounce arrow: Bounces off the pumpkin back toward the archer, while the pumpkin swings to angle θ_bounce.
4. Rank the three angles θ_emb, θ_pass, and θ_bounce from greatest to least in the spaces indicated below. Use “1” for the greatest angle, “2” for the next greatest, and so on. If any two or all three angles are the same, use the same number for their ranking.
5. ______θ_emb ______θ_pass _______θ_bounce
6. In a clear, coherent, paragraph-length response that may also contain figures and/or equations, justify your ranking.

Answer 1

Hi Matthew O.,

Momentum transfer is the key. If the arrow bounces backward, it now has negative momentum. Thus, it transferred all its initial positive momentum to the pumpkin, and then some! The embedded arrow transferred almost all its momentum to the pumpkin, keeping only enough to travel with the pumpkin in its new forward speed; the penetrating and exiting arrow didn't lose all its initial momentum. So the speeds after collision is over for the three pumpkins are in decreasing magnitude, hence also the angles swung up to.

Incidentally, I would take issue with the description of the three arrowheads as being "of different materials". They might well be, but that's not germane to their performance. What would matter would be the area presented perpendicularly to the direction of arrow travel, i.e. their design. Hunting arrows have 3 razor-sharp sloped metal edges that meet at the tip in a sharp point. If you could ask them if they could pass through a pumpkin, and they might reply, "How many?"