Tutorials In Introductory Physics Homework Solutions Conservation Of Momentum In One Dimension

Tutorials in Introductory Physics Homework Solutions Conservation of Momentum in One Dimension

Conservation of momentum is one of the fundamental principles of physics. It states that the total momentum of a system of objects remains constant, as long as no external forces act on the system. In other words, the momentum of a system can only change if there is a net force acting on it. In this article, we will explore some homework problems from the book Tutorials in Introductory Physics by Lillian C. McDermott and Peter S. Shaffer, and learn how to apply the conservation of momentum principle to solve them.

Problem 1: Collision on an air track

In this problem, two carts of different masses are moving on a frictionless air track. Cart 1 has a mass of 0.5 kg and an initial velocity of 2 m/s to the right. Cart 2 has a mass of 1 kg and an initial velocity of -1 m/s to the left. The carts collide and stick together. What is the final velocity of the combined cart?

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To solve this problem, we need to use the conservation of momentum principle. The initial momentum of the system is equal to the final momentum of the system, since there are no external forces acting on it. We can write this as:

$$m_1v_1 + m_2v_2 = (m_1 + m_2)v_f$$

where $m_1$ and $m_2$ are the masses of cart 1 and cart 2, respectively, $v_1$ and $v_2$ are their initial velocities, and $v_f$ is their final velocity after the collision. Plugging in the given values, we get:

$$0.5 \times 2 + 1 \times (-1) = (0.5 + 1)v_f$$

Solving for $v_f$, we get:

$$v_f = \frac0.5 \times 2 + 1 \times (-1)0.5 + 1 = -\frac13 \text m/s$$

Therefore, the final velocity of the combined cart is -0.33 m/s to the left.

Problem 2: Explosion on an air track

In this problem, a cart with a mass of 0.8 kg is initially at rest on a frictionless air track. The cart contains an explosive charge that splits it into two pieces: one with a mass of 0.3 kg and one with a mass of 0.5 kg. The piece with a mass of 0.3 kg moves to the right with a velocity of 4 m/s after the explosion. What is the velocity of the other piece?

To solve this problem, we need to use the conservation of momentum principle again. The initial momentum of the system is zero, since the cart is at rest before the explosion. The final momentum of the system is equal to the sum of the momenta of the two pieces after the explosion. We can write this as:

$$0 = m_3v_3 + m_4v_4$$

where $m_3$ and $m_4$ are the masses of the two pieces, respectively, and $v_3$ and $v_4$ are their velocities after the explosion. Plugging in the given values, we get:

$$0 = 0.3 \times 4 + 0.5 \times v_4$$

Solving for $v_4$, we get:

$$v_4 = -\frac0.3 \times 40.5 = -2.4 \text m/s$$

Therefore, the velocity of the other piece is -2.4 m/s to the left.

Conclusion

In this article, we have learned how to use the conservation of momentum principle to solve some homework problems from the book Tutorials in Introductory Physics. We have seen that this principle applies to both collisions and explosions, and that it allows us to find the final velocities of the objects involved, given their initial velocities and masses. We have also learned how to use algebra and arithmetic to manipulate the equations and solve for the unknowns. We hope that this article has helped you understand and apply the conservation of momentum principle better.

For more information and practice problems on conservation of momentum, you can check out the following resources: