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 (are) 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 total momentum of the system decreases as the fragments spread out. 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 will change its trajectory when the first fragment touches the ground. e) The trajectory of the center of mass depends on the number of fragments and their velocities right after the explosion. Answer: b) and c) are the true statements.

Step-by-step solution

Statement a: Exploding shell and momentum conservation

The force of the explosion is an internal force, as it happens within the system (the fragments). An internal force does not change the total momentum of the system, and thus the momentum of the system is conserved during the explosion. Therefore, statement a) is false.

Statement b: Internal force and total momentum

As explained in the analysis of statement a), the force of the explosion is an internal force, which cannot alter the total momentum of the system. So, statement b) is correct and true.

Statement c: Center of mass after the explosion

The center of mass of the fragments will continue to move along the initial parabolic trajectory, as the total momentum of the system is conserved. It does not matter when the last fragment touches the ground - the center of mass will continue on that trajectory until then. Hence, statement c) is true.

Statement d: Center of mass and first fragment touchdown

Similar to the explanation in statement c), the center of mass will continue to move along the initial parabolic trajectory regardless of when the first fragment touches the ground. Therefore, statement d) is incorrect.

Statement e: Trajectory of the center of mass and fragment properties

Since the total momentum of the system is conserved and the internal force of the explosion does not alter it, the trajectory of the center of mass does not depend on the number of fragments or their velocities right after the explosion. It will continue along the initial parabolic trajectory. So, statement e) is false. In conclusion, the true statements are: b) The force of the explosion is an internal force and thus cannot alter the total momentum of the system and 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.

Key concepts

Internal Forces in Physics

Understanding internal forces is crucial when studying systems in physics. An internal force refers to any force that acts between the elements within a system. To visualize this, imagine a row of dancers holding hands; if they push or pull each other, they exert internal forces on one another. In contrast, external forces come from outside the system, like a wind pushing the dancers.

When considering an object or a system of objects—like the fragments of the exploded artillery shell in our exercise—the concept of internal force tells us that these forces do not change the total momentum of the system. Why is that? It's because according to Newton's third law, for every action, there is an equal and opposite reaction. Within a closed system (like the shell and its fragments after explosion), the internal forces cancel each other out, effectively leading to no change in overall momentum. Therefore, the explosion of the shell, even though dramatic, is an internal event and does not affect the total momentum of the fragments collectively.

Center of Mass

The center of mass is a point that represents the average position of all the mass in a system. Think of it as a balancing point. If you had a physical model of the fragmented shell, the center of mass is the point where you could balance the model perfectly on the tip of your finger.

During the explosion of our artillery shell, even though the individual fragments may fly off in different directions with varying speeds, the center of mass of the system maintains its trajectory. This is due to the conservation of momentum. In essence, if there are no external forces acting on the system (ignoring air resistance), the center of mass behaves as though the explosion never happened, following the same parabolic path that the shell was on before it exploded. It's a potent concept that applies to various systems, from understanding the movement of celestial bodies to the dynamics of sports.

Trajectory in Physics

The trajectory of an object is the path that it follows as it moves through space. It's shaped by the initial conditions of the object's movement and the forces that act upon it—both internal and external. In the case of projectile motion, such as an artillery shell being fired and then moving in a parabolic path, gravity is the primary external force at work.

Once the shell explodes and breaks into fragments, the individual pieces may scatter in many directions, but what's fascinating is that the collective center of mass of these pieces maintains the original trajectory. This is a grand display of the robustness of conserved quantities like momentum in the face of dynamic events. By understanding the nature of trajectories, we can predict the paths of satellites, plan interplanetary missions, or analyze the performance of athletes in sports like basketball or soccer, where projectile motion is a common occurrence.