Two Figure Skaters Are Performing. The Man (75 Kg) Travels At 1.2 M/s 1.2 \, \text{m/s} 1.2 M/s And The Woman (60 Kg) Jumps Towards Him At − 3.1 M/s -3.1 \, \text{m/s} − 3.1 M/s . When The Man Catches The Woman, What Will Their Speed Be?$[ (75 , \text{kg}

Introduction

In this article, we will explore a physics problem involving two figure skaters performing on ice. The problem requires us to calculate the speed of the two skaters when the man catches the woman. We will use the principles of conservation of momentum to solve this problem.

The Problem

Two figure skaters are performing on ice. The man weighs 75 kg and is traveling at a speed of $1.2 \, \text{m/s}$. The woman weighs 60 kg and is jumping towards the man at a speed of $-3.1 \, \text{m/s}$. When the man catches the woman, what will their speed be?

Conservation of Momentum

The law of conservation of momentum states that the total momentum of a closed system remains constant over time. In this case, the system consists of the man and the woman. We can write the equation for conservation of momentum as follows:

$$m_1 \vec{v}_1 + m_2 \vec{v}_2 = (m_1 + m_2) \vec{v}_{\text{final}}$$

where $m_1$ and $m_2$ are the masses of the man and the woman, respectively, $\vec{v}_1$ and $\vec{v}_2$ are their initial velocities, and $\vec{v}_{\text{final}}$ is their final velocity.

Calculating the Final Velocity

We can plug in the values given in the problem into the equation for conservation of momentum:

$$75 \, \text{kg} \cdot 1.2 \, \text{m/s} + 60 \, \text{kg} \cdot (-3.1 \, \text{m/s}) = (75 \, \text{kg} + 60 \, \text{kg}) \vec{v}_{\text{final}}$$

Simplifying the equation, we get:

$$90 \, \text{kg} \cdot \text{m/s} - 186 \, \text{kg} \cdot \text{m/s} = 135 \, \text{kg} \cdot \vec{v}_{\text{final}}$$

Combining like terms, we get:

$$-96 \, \text{kg} \cdot \text{m/s} = 135 \, \text{kg} \cdot \vec{v}_{\text{final}}$$

Dividing both sides by $135 \, \text{kg}$, we get:

$$\vec{v}_{\text{final}} = -\frac{96 \, \text{kg} \cdot \text{m/s}}{135 \, \text{kg}}$$

Simplifying the expression, we get:

$$\vec{v}_{\text{final}} = -0.71 \, \text{m/s}$$

Conclusion

In this article, we used the principle of conservation of momentum to solve a physics problem involving two figure skaters performing on ice. We calculated the final velocity of the two skaters when the man catches the woman, and found that it is $-0.71 \, \text{m/s}$.

References

• Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
• Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Additional Resources

• Khan Academy: Conservation of Momentum
• Physics Classroom: Conservation of Momentum
• MIT OpenCourseWare: Physics 8.01: Classical Mechanics

Discussion

What do you think about this problem? Do you have any questions or comments? Please feel free to discuss in the comments section below.

Related Problems

• A 50 kg box is moving at a speed of $2 \, \text{m/s}$. A 20 kg box is moving towards it at a speed of $-3 \, \text{m/s}$. What is the final velocity of the two boxes when they collide?
• A 75 kg skater is moving at a speed of $1.2 \, \text{m/s}$. A 60 kg skater is jumping towards him at a speed of $-3.1 \, \text{m/s}$. What is the final velocity of the two skaters when the man catches the woman?

Glossary

• Conservation of momentum: The law that states that the total momentum of a closed system remains constant over time.
• Momentum: The product of an object's mass and velocity.
• Velocity: The rate of change of an object's position with respect to time.
  Q&A: Two Figure Skaters Performing =====================================

Q: What is the law of conservation of momentum?

A: The law of conservation of momentum states that the total momentum of a closed system remains constant over time. In other words, the total momentum before a collision or interaction is equal to the total momentum after the collision or interaction.

Q: How is momentum calculated?

A: Momentum is calculated by multiplying an object's mass by its velocity. The formula for momentum is:

$$\vec{p} = m \vec{v}$$

where $\vec{p}$ is the momentum, $m$ is the mass, and $\vec{v}$ is the velocity.

Q: What is the difference between velocity and momentum?

A: Velocity is the rate of change of an object's position with respect to time, while momentum is the product of an object's mass and velocity. In other words, velocity tells us how fast an object is moving, while momentum tells us how much "oomph" it has.

Q: Can you give an example of how to use the law of conservation of momentum?

A: Let's go back to the problem of the two figure skaters. We can use the law of conservation of momentum to calculate the final velocity of the two skaters when the man catches the woman. We can write the equation for conservation of momentum as follows:

$$m_1 \vec{v}_1 + m_2 \vec{v}_2 = (m_1 + m_2) \vec{v}_{\text{final}}$$

where $m_1$ and $m_2$ are the masses of the man and the woman, respectively, $\vec{v}_1$ and $\vec{v}_2$ are their initial velocities, and $\vec{v}_{\text{final}}$ is their final velocity.

Q: What is the final velocity of the two skaters when the man catches the woman?

A: We can plug in the values given in the problem into the equation for conservation of momentum:

$$75 \, \text{kg} \cdot 1.2 \, \text{m/s} + 60 \, \text{kg} \cdot (-3.1 \, \text{m/s}) = (75 \, \text{kg} + 60 \, \text{kg}) \vec{v}_{\text{final}}$$

Simplifying the equation, we get:

$$90 \, \text{kg} \cdot \text{m/s} - 186 \, \text{kg} \cdot \text{m/s} = 135 \, \text{kg} \cdot \vec{v}_{\text{final}}$$

Combining like terms, we get:

$$-96 \, \text{kg} \cdot \text{m/s} = 135 \, \text{kg} \cdot \vec{v}_{\text{final}}$$

Dividing both sides by $135 \, \text{kg}$, we get:

$$\vec{v}_{\text{final}} = -\frac{96 \, \text{kg} \cdot \text{m/s}}{135 \, \text{kg}}$$

Simplifying the expression, we get:

$$\vec{v}_{\text{final}} = -0.71 \, \text{m/s}$$

Q: What is the significance of the negative sign in the final velocity?

A: The negative sign in the final velocity indicates that the two skaters are moving in opposite directions. In other words, the man is moving to the right, while the woman is moving to the left.

Q: Can you give another example of how to use the law of conservation of momentum?

A: Let's say we have a 50 kg box that is moving at a speed of $2 \, \text{m/s}$. A 20 kg box is moving towards it at a speed of $-3 \, \text{m/s}$. We can use the law of conservation of momentum to calculate the final velocity of the two boxes when they collide.

We can write the equation for conservation of momentum as follows:

$$m_1 \vec{v}_1 + m_2 \vec{v}_2 = (m_1 + m_2) \vec{v}_{\text{final}}$$

where $m_1$ and $m_2$ are the masses of the two boxes, respectively, $\vec{v}_1$ and $\vec{v}_2$ are their initial velocities, and $\vec{v}_{\text{final}}$ is their final velocity.

We can plug in the values given in the problem into the equation for conservation of momentum:

$$50 \, \text{kg} \cdot 2 \, \text{m/s} + 20 \, \text{kg} \cdot (-3 \, \text{m/s}) = (50 \, \text{kg} + 20 \, \text{kg}) \vec{v}_{\text{final}}$$

Simplifying the equation, we get:

$$100 \, \text{kg} \cdot \text{m/s} - 60 \, \text{kg} \cdot \text{m/s} = 70 \, \text{kg} \cdot \vec{v}_{\text{final}}$$

Combining like terms, we get:

$$40 \, \text{kg} \cdot \text{m/s} = 70 \, \text{kg} \cdot \vec{v}_{\text{final}}$$

Dividing both sides by $70 \, \text{kg}$, we get:

$$\vec{v}_{\text{final}} = \frac{40 \, \text{kg} \cdot \text{m/s}}{70 \, \text{kg}}$$

Simplifying the expression, we get:

$$\vec{v}_{\text{final}} = 0.57 \, \text{m/s}$$

Q: What is the significance of the positive sign in the final velocity?

A: The positive sign in the final velocity indicates that the two boxes are moving in the same direction. In other words, both boxes are moving to the right.

Conclusion

In this article, we have discussed the law of conservation of momentum and how it can be used to solve problems involving collisions and interactions. We have also given examples of how to use the law of conservation of momentum to calculate the final velocity of objects in different scenarios.