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
vo toward the right.
The arrowheads are made of different materials, however, and each
behaves differently when it strikes the pumpkin, as described below.
 Embedded arrow: Strikes the pumpkin and remains embedded,
while the pumpkin swings to angle θemb
 Pass arrow: Passes all the way through the pumpkin and continues traveling away from the archer, while the
pumpkin swings to angle θpass
 Bounce arrow: Bounces off the pumpkin back toward the archer, while the pumpkin swings to angle θbounce
(a) Rank the three angles θemb, θpass, θbounce 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.

(b) In a clear, coherent, paragraph‐length response that may also contain figures and/or equations, justify your ranking

a.  The greatest angle will be bouncing, then embedded and finally the least is passing angle.

b. The  Momentum is greatest when the arrow bounces and least when the arrow passes.  This justify the obtained ranking of angles.

The given problem is based on the concept of momentum and angle made at various instance. Momentum is defined as an impact caused due to applied force. It is expressed as the product of mass and velocity of an object, with respect to the direction. Taking forward direction of arrow as positive then backward direction will be negative.

From the law of conservation of linear momentum, the sum of initial and final momentum should be equal.

Thus we can conclude that by ranking the angles, consider that the maximum angle that the pumpkin reaches depends on the speed of the pumpkin. The greatest angle will be bouncing, then embedded and finally the least is passing angle.

Learn more about the conservation of momentum here:

link

1. Bounce arrow, embedded arrow and pass arrow

2. Momentum is greatest when the arrow bounces and least when the arrow passes.

Explanation:

Momentum is the product of mass and velocity of an object, with respect to the direction. Taking forward direction of arrow as positive then backward direction will be negative.

From the law of conservation of linear momentum, the sum of initial and final momentum should be equal.

When the arrow moves forward then bounces, it means the pumpkin will have positive momentum while arrow during bouncing will have negative energy.

When the arrow is embedded in the pumpkin, both pumpkin and arrow will have same momentum

Finally, when the arrow passes through the pumpkin, it means it will have positive momentum hence pumpkin momentum is very little.

Therefore, by ranking the angles, we consider that the maximum angle that the pumpkin reaches depends on the speed of the pumpkin. The greatest angle will be bouncing, then embedded and finally the least is passing angle.

kepler used the data from tycho blahe (his mentor) to construct the laws of planetary motion.