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Chapter 7: Problem 55

A \(5.0-\mathrm{kg}\) howler monkey is swinging due east on a vine. It overtakes and grabs onto a \(6.0-\mathrm{kg}\) monkey also moving east on a second vine. The first monkey is moving at \(12 \mathrm{~m} / \mathrm{s}\) at the instant it grabs the second, which is moving at \(8 \mathrm{~m} / \mathrm{s}\). After they join on the same vine, what is their common speed? SSM

Short Answer

Expert verified
The common speed of the two monkeys after they grab onto each other is calculated by dividing the total momentum by the combined mass of the two monkeys.

Step by step solution

01

Compute the initial momentum

Momentum is defined as the product of an object's mass and its velocity. The initial momentum before the monkeys grab onto each other can be calculated by adding the individual momentum of each monkey. It is calculated as follows: Initial momentum = (mass of first monkey * velocity of first monkey) + (mass of second monkey * velocity of second monkey) = \( (5.0 kg * 12 m/s) + (6.0 kg * 8 m/s) \)
02

Compute the final momentum

After the monkeys grab onto each other, they will swing together with the same velocity. Since there are no external forces, the total momentum will be conserved. The final momentum will be the same as the initial momentum we calculated in Step 1, which is the combined mass of the two monkeys times their common speed. It is calculated as follows: Final momentum = initial momentum = (combined mass * common speed)
03

Solve for the common speed

We can find the common speed by dividing the total momentum by the combined mass of the two monkeys: Common speed = final momentum / combined mass. Substitute the known values, solve for the common speed, and obtain the answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physics Problem
In this scenario, we have two monkeys swinging on vines in the same direction, and we need to determine their speed once they join together. This is a classic physics problem dealing with the concept of momentum.
Understanding how objects interact in motion and how their momentum changes when they collide or join is core to solving problems like these.
In our case:
  • We have a 5 kg monkey swinging at 12 m/s.
  • Another 6 kg monkey is swinging at 8 m/s.
Our goal is to find out their shared speed after they hold onto each other, creating a single moving system. This problem will use the law of conservation of momentum to find the solution.
Momentum Calculation
Momentum is a fundamental concept in physics, defined as the product of an object's mass and velocity. It helps describe how much motion an object has and is a vector quantity, meaning it has both magnitude and direction.
In our monkey problem, calculating the initial momentum is the key starting point. For each monkey, momentum can be calculated as:
  • First monkey: \( (5.0 \, \text{kg}) \times (12 \, \text{m/s}) = 60 \, \text{kg} \cdot \text{m/s} \)
  • Second monkey: \( (6.0 \, \text{kg}) \times (8 \, \text{m/s}) = 48 \, \text{kg} \cdot \text{m/s} \)
Total initial momentum is then simply the sum of these individual momenta: \[ 60 \, \text{kg} \cdot \text{m/s} + 48 \, \text{kg} \cdot \text{m/s} = 108 \, \text{kg} \cdot \text{m/s} \]This calculation tells us the total amount of motion the system has before the monkeys join together.
Inelastic Collision
When the monkeys grab onto each other, they no longer move independently, exemplifying an inelastic collision. In inelastic collisions, colliding objects stick together, and kinetic energy is not conserved. However, momentum is always conserved.
By applying the conservation of momentum principle, the final momentum of the system (after they grab each other) equals the initial momentum.
Here鈥檚 what happens:
  • The combined mass is the sum of their masses: \( 5.0 \, \text{kg} + 6.0 \, \text{kg} = 11.0 \, \text{kg} \).
  • Since the final momentum equals the initial momentum, we use: \( \text{Common speed} = \frac{\text{Initial Momentum}}{\text{Combined Mass}} \).
  • Substitute: \( \frac{108 \, \text{kg} \cdot \text{m/s}}{11.0 \, \text{kg}} \= 9.82 \, \text{m/s} \).
Thus, the common speed of the two monkeys after the inelastic collision is approximately 9.82 m/s.

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