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Chapter 8: Problem 13

A projectile is launched into the air. Part way through its flight, it explodes. How does the explosion affect the motion of the center of mass of the projectile?

Short Answer

Expert verified
Answer: The explosion does not affect the motion of the center of mass of the projectile in mid-flight, as the total momentum of the system is conserved and there are no net external forces acting on it after the explosion. The center of mass continues along its original trajectory with the same velocity.

Step by step solution

01

Understanding the principle of conservation of momentum

The principle of conservation of momentum states that for a closed system, the total momentum is conserved if there are no net external forces acting on it. In the case of the exploding projectile, if we consider the projectile and the fragments after the explosion as part of the same system, there will be no net external force acting on the system.
02

Establish the initial conditions

Before the explosion, the projectile is in mid-flight, and we can define its initial mass as M and initial velocity as v. The initial momentum of the projectile can be calculated as: momentum_initial = Mv
03

Consider the moment of explosion

At the moment of explosion, the projectile fragments into multiple pieces. Let's assume the total mass of the fragments remains M. Each fragment will have a different velocity due to the explosion, and the overall momentum of the fragments will be the vector sum of the momenta of individual fragments.
04

Applying the principle of conservation of momentum

Since there is no net external force acting on the projectile, the total momentum of the projectile remains conserved after the explosion. Therefore, the momentum of the fragments must equal the initial momentum of the projectile: momentum_fragments = Mv
05

Analyzing the motion of the center of mass

The center of mass of a system moves along a trajectory determined by the external forces acting on it, ignoring internal forces. In the case of the exploding projectile, there are no net external forces acting on the system after the explosion. Therefore, the motion of the center of mass of the projectile after the explosion will continue along its original trajectory, with the same velocity. The explosion will not affect the motion of the center of mass of the projectile. In conclusion, the motion of the center of mass of the projectile will remain unaffected by the explosion, as the total momentum of the system is conserved and there are no net external forces acting on the system after the explosion.

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Most popular questions from this chapter

Starting at rest, two students stand on 10.0 -kg sleds, which point away from each other on ice, and they pass a 5.00 -kg medicine ball back and forth. The student on the left has a mass of \(50.0 \mathrm{~kg}\) and can throw the ball with a relative speed of \(10.0 \mathrm{~m} / \mathrm{s}\). The student on the right has a mass of \(45.0 \mathrm{~kg}\) and can throw the ball with a relative speed of \(12.0 \mathrm{~m} / \mathrm{s} .\) (Assume there is no friction between the ice and the sleds and no air resistance.) a) If the student on the left throws the ball horizontally to the student on the right, how fast is the student on the left moving right after the throw? b) How fast is the student on the right moving right after catching the ball? c) If the student on the right passes the ball back, how fast will the student on the left be moving after catching the pass from the student on the right? (d) How fast is the student on the right moving after the pass?

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