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 of a projectile in mid-air does not affect the motion of its center of mass. The center of mass will continue to follow the original parabolic path as if the explosion had not occurred.

Step-by-step solution

Understand the properties of the center of mass

The center of mass of a projectile is the point at which the entire mass of the projectile may be assumed to be concentrated. The motion of the center of mass is only affected by external forces acting on the system.

Apply Newton’s second law to the center of mass

Newton's second law states that the acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass. We can write the equation for the center of mass as: m a_{CM} = \sum F_ext Here, a_{CM} represents the acceleration of the center of mass, m is the total mass of the projectile, and \sum F_ext represents the sum of the external forces acting on the system. Before the explosion, the only external force acting on the projectile is gravity. Thus, the equation for the center of mass becomes: m a_{CM,initial} = -m g

Consider the conservation of momentum during the explosion

When the projectile explodes, the momentum of its fragments should be conserved because the explosion is an internal force, and internal forces do not change the total momentum of the system. Hence, the momentum of the center of mass before the explosion will be equal to the momentum of the center of mass after the explosion.

Analyze the motion of the center of mass after the explosion

After the explosion, the only external force acting on the center of mass is still gravity. So, the equation for the center of mass after the explosion remains the same: m a_{CM,final} = -m g Notice that the acceleration of the center of mass before and after the explosion is the same (-m g).

Conclusion

Since the external forces acting on the system remain unchanged, and the momentum of the center of mass is conserved during the explosion, the motion of the center of mass is not affected by the explosion. In other words, the projectile's center of mass will continue to follow the original parabolic path as if the explosion had not occurred.

Key concepts

Center of Mass

The center of mass (CM) is a fundamental concept in physics, especially when analyzing the motion of objects. It represents a point where the total mass of an object or system of objects can be thought of as being concentrated. When a projectile is launched into the air, its center of mass follows a predictable path, typically a parabola due to the force of gravity.

In the case of the projectile's explosion midair presented in the exercise, the CM is crucial for predicting the subsequent motion of the fragments. Despite the explosion being a dramatic internal event, the CM of the projectile system will continue to move as if the explosion never occurred. This can be quite counterintuitive, but it illustrates the importance of the CM in simplifying complex physical problems into more manageable analyses. The overall path of the CM is influenced only by external forces like gravity, not by the internal forces such as those present in an explosion.

Newton's Second Law

Newton's second law of motion is one of the cornerstones of classical mechanics and states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (\( F = m \times a \)). For our projectile motion scenario, Newton's second law helps us determine how external forces affect the center of mass.

Before the explosion, gravity is the sole external force acting on the projectile, resulting in a downward acceleration. Newton's second law allows us to quantify this acceleration as it relates to the mass of the projectile and the strength of gravity (represented by the gravitational constant, g). After the explosion, assuming no other external forces intervene, the second law confirms that the acceleration due to gravity will continue to act on the center of mass in the same manner, thus the motion of the CM will remain unaffected by the internal forces of the explosion.

Conservation of Momentum

Momentum is conserved in the absence of external forces, as described by the law of conservation of momentum. This fundamental principle tells us that if no external forces act on a system, its total momentum cannot change. In the context of the exploding projectile, even though the projectile is breaking into pieces, since the forces causing it to explode are internal, the total momentum of the projectile fragments right after the explosion must equal the total momentum immediately before the explosion.

This principle provides a powerful tool for solving problems in physics, as it enables us to predict the subsequent motion of objects after a collision or explosion. In our exercise, it assures us that the center of mass for the projectile will continue to move with the same velocity in the horizontal direction, even after the explosion — the internal forces of the explosion have no power to alter the horizontal component of the momentum.

External Forces

External forces are forces that stem from outside a system and can change the system's total momentum. Typical examples include gravitational pull, friction, or a push or pull from an object not part of the system. In projectile motion, the primary external force is gravity. It acts continuously on the projectile's center of mass and determines its trajectory.

Distinguishing between internal and external forces is crucial when analyzing physical situations. Internal forces, such as the explosive forces in our projectile example, can change the shape or distribution of mass within a system but do not affect the overall motion of the system's center of mass. The motion of the center of mass of our projectile post-explosion is a perfect demonstration of this principle; despite the internal upheaval, the path of the center of mass remains uninterrupted because no new external forces have been introduced.