Question

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

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

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.

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

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.

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

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.