Question

An artillery shell is moving on a parabolic trajectory when it explodes in midair. The shell shatters into a very large number of fragments. Which of the following statements is true (select all that apply)? a) The force of the explosion will increase the momentum of the system of fragments, and so the momentum of the shell is not conserved during the explosion. b) The force of the explosion is an internal force and thus cannot alter the total momentum of the system. c) The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the last fragment touches the ground. d) The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the first fragment touches the ground. e) The center of mass of the system of fragments will have a trajectory that depends on the number of fragments and their velocities right after the explosion.

Short answer

a) The force of the explosion will increase the momentum of the system of fragments, and so the momentum of the shell is not conserved during the explosion. b) The force of the explosion is an internal force and thus cannot alter the total momentum of the system. c) The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the last fragment touches the ground. d) The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the first fragment touches the ground. e) The center of mass of the system of fragments will have a trajectory that depends on the number of fragments and their velocities right after the explosion. Answer: The correct statements are b) and c).

Step-by-step solution

Statement a

The force of the explosion will increase the momentum of the system of fragments, and so the momentum of the shell is not conserved during the explosion. This statement is false. The total momentum of a closed system is always conserved. The force of the explosion is an internal force, and internal forces cannot change the total momentum of the system.

Statement b

The force of the explosion is an internal force and thus cannot alter the total momentum of the system. This statement is true. As mentioned before, the forces involved in the explosion are internal forces. These forces can change the individual momenta of the fragments but not the total momentum of the system.

Statement c

The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the last fragment touches the ground. This statement is true. The center of mass of a system of particles moves as if all the mass of the system were concentrated at that point. Since the total momentum of the system is conserved during the explosion, the center of mass will continue moving along the initial parabolic trajectory until the last fragment touches the ground.

Statement d

The center of mass of the system of fragments will continue to move on the initial parabolic trajectory until the first fragment touches the ground. This statement is false. As explained in the analysis of statement c, the center of mass will continue moving along the initial parabolic trajectory until the last fragment touches the ground, not the first.

Statement e

The center of mass of the system of fragments will have a trajectory that depends on the number of fragments and their velocities right after the explosion. This statement is false. The trajectory of the center of mass only depends on the initial parabolic trajectory and does not depend on the number of fragments or their individual velocities. In conclusion, the statements b and c are true.

Key concepts

Internal Forces

When an object like an artillery shell explodes midair, it breaks into many pieces due to the force of the explosion. However, this force is classified as an **internal force**. Internal forces occur within a system of particles and can include explosions or electrical interactions within the object itself. Importantly, internal forces do not affect the total momentum of the system.

This means that even though individual pieces of the shell might fly off in different directions with various speeds, the overall momentum of the entire system remains unchanged. This concept is a key aspect of the conservation of momentum. If the total external forces are zero, the momentum before and after the explosion stays the same.

So, remember that internal forces can't change the total momentum. They only rearrange how momentum is distributed among the parts in the system.

Center of Mass

The **center of mass** of an object or system is a point where all of its mass might be thought of as being concentrated. When it comes to a system of fragments like an exploding shell, the center of mass plays a critical role in understanding motion.

After the explosion, each fragment follows its own path. Yet, the center of mass of the entire system continues on the same path it was following before the explosion. This means that if the shell was on a parabolic trajectory, the center of mass keeps moving along that initial path. It doesn't matter where individual fragments land first; the center of mass motion aligns with the trajectory before any piece hits the ground.

This constant path is due to the conservation of momentum and the fact that there are no external forces acting on the system, apart from gravity, which acts equally on all parts.

Parabolic Trajectory

A **parabolic trajectory** is the curved path that objects follow under the influence of gravity alone, assuming no air resistance. This type of motion is observed in projectiles, such as an artillery shell following a curved arc through the air.

When the shell undergoes an explosion mid-flight, each fragment will commence its own motion based on the explosion's dynamics. However, the center of mass for all these fragments will continue to move along the original parabolic path as if the shell had not exploded.

This happens because the momentum of the system is conserved and gravity, being a consistent force, continues to treat the shell fragments as if they were still one object. This fascinating behavior ensures that the overall path of the original system's center of mass remains unaffected by how the pieces separate, focusing instead on the initial trajectory.