Momentum, Collisions and Explosions Legacy Problem #14 Guided Solution
Problem*
A candy-filled piñata is hung from a tree for Matthew's birthday. During an unsuccessful attempt to break the 4.4-kg piñata, Hayden cracks it with a 0.54-kg stick moving at 4.8 m/s. The stick stops and the piñata undergoes a gentle swinging motion. Determine the swing speed of the piñata immediately after being cracked by the stick.
Audio Guided Solution
This problem statement gives us a picture of a piñata hanging at rest from a tree, while a boy comes along with a stick and swings at the piñata. There is a collision between the stick and the at-rest piñata. The stick changes its velocity from 4.8 meters per second to 0 meters per second. The mass of the stick is given. This momentum change of the stick is accompanied by a momentum change of the piñata, and we wish to find the speed of the piñata immediately after being cracked by the stick. The way we'll do this is we'll use the idea that the momentum change of one object in a collision is equal in magnitude and opposite in direction to the momentum change of the other object. Since we know the stick's mass and its original and final speed, we can find its momentum change. Once we find the momentum change of the stick, we can set it equal to the mass times the velocity change of the piñata, and we can find the velocity change of the piñata. Once we know that, we can answer the question, what's the swing speed of the piñata immediately after the crack with the stick? So, here we go. We're going to begin by finding the momentum change of the stick. It was moving forward at 4.8 meters per second. It hit the piñata and it slowed down to 0 meters per second. So its velocity change is negative 4.8 meters per second. Multiplying this by 0.54 kilograms gives us a momentum change of that stick. It's 2.592 kilograms times a meter per second, and it's a negative value. Now, that means that the piñata gains momentum if the stick lost it. And so, we can say this 2.592 is equal to the momentum change of the piñata, or is equal to the mass of the piñata times the velocity change of the piñata. The mass of the piñata is 4.4, so I can put that into the equation, and then divide through by 4.4. So the 2.592 kilograms meters per second divided by the 4.4 kilograms gives me a velocity change of the piñata. It comes out to be about 0.589 meters per second, or rounding to two digits, 0.59 meters per second. That's the velocity change, and since it started from rest, that's also the speed immediately after being cracked by the stick.
Solution
0.59 m/s
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities in an organized manner. Equate given values to the symbols used to represent the corresponding quantity - e.g., \(m = 1.50 \unit{kg}\), \(v_i = 2.68 \unit{\meter\per\second}\), \(F = 4.98 \unit{\newton}\), \(t = 0.133 \unit{\second}\), \(v_f = \colorbox{gray}{Unknown}\).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
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