Question

Two cannonballs are shot in sequence from a cannon, into the air, with the same muzzle velocity, at the same launch angle. Based on their trajectory and range, how can you tell which one is made of lead and which one is made of wood. If the same cannonballs where launched in vacuum, what would the answer be?

Short answer

Answer: The lead cannonball would have a longer range and a trajectory closer to the ideal parabolic motion due to its greater mass and density, which results in a smaller deceleration relative to its mass due to air resistance compared to the lighter wooden cannonball.

Step-by-step solution

Analyze the forces acting on the cannonballs

When the cannonballs are launched, the primary forces acting on them are gravity and air resistance. Since they have the same launch angle and velocity, we can differentiate between the two based on how much air resistance affects their motion. As lead has greater mass and density, it will experience a smaller deceleration relative to its mass due to air resistance compared to the lighter wooden cannonball.

Determine the effect of air resistance on the trajectory

The air resistance will have a greater impact on the trajectory of the lighter wooden cannonball than on the heavier lead one, causing it to have a shorter range and a trajectory that differs from the ideal parabolic path more than the lead cannonball would. Thus, the wooden cannonball's trajectory will deviate more from the ideal parabolic motion compared to the lead cannonball.

Compare the trajectory and range

When comparing the trajectory and range of the two cannonballs, the one made of lead will have a longer range and a trajectory closer to the ideal parabolic motion. The wooden cannonball will have a shorter range and a trajectory that deviates more from the ideal parabolic path. This is due to the larger effect of air resistance on the lighter wooden cannonball.

Determine the result in vacuum

If the same cannonballs were launched in a vacuum, there would be no air resistance to act on them. Therefore, both cannonballs would follow an ideal parabolic trajectory, and their range would only be determined by their initial launch angle and velocity. Since both launch angle and velocity are identical for the two cannonballs, they would have the same trajectory and range in a vacuum. So you would not be able to tell which one is made of lead and which one is made of wood based on their trajectory and range in a vacuum.

Key concepts

Air Resistance

When an object moves through the air, it encounters a force known as air resistance. This force acts in the opposite direction of the object's motion, making it slow down over time. The amount of air resistance depends on several factors:

• The object's speed: Faster objects experience more air resistance.
• The surface area: Larger surfaces catch more air, leading to greater resistance.
• The object's shape: Streamlined shapes experience less drag.
• The density of the air: Denser air increases the resistance.

In the context of projectile motion, heavier objects like lead cannonballs are less affected by air resistance compared to lighter ones like wooden cannonballs. This is because gravity exerts a stronger force on the heavier object, allowing it to maintain its speed better, resulting in a lesser alteration of its trajectory due to air resistance. This means the lead cannonball can travel further and more predictably than the wooden one under the same conditions.

Parabolic Trajectory

A projectile, when launched at an angle, ideally follows a curved path known as a parabolic trajectory. This path is shaped like a parabola due to the constant acceleration of gravity acting on the projectile in a vertical direction, while its initial velocity propels it horizontally. Several factors influence how closely a real-world object's path resembles a parabola:

• Initial speed and angle: These determine the shape and length of the trajectory.
• Air resistance: This can cause the parabolic path to become distorted and shorter.
• Mass of the projectile: Heavier objects are less affected by air resistance and thus maintain a more parabolic motion.

In practice, the presence of air resistance causes the path of a lighter object, like a wooden cannonball, to deviate more from this ideal parabola compared to a heavier lead cannonball, leading to a noticeable difference in their ranges and paths.

Vacuum

A vacuum is a space devoid of air and, consequently, air resistance. In such an environment, an object's trajectory is affected solely by gravity, not by air drag. Here's why launching cannonballs in a vacuum changes their motion:

• Objects, regardless of mass, only experience gravitational force, hence their trajectories are solely determinate by gravity.
• Both a wooden and a lead cannonball will follow the same path, as no air resistance is present to alter the trajectory based on mass or surface area.
• The parabolic trajectory remains pure and undistorted since air drag is non-existent.

Without air resistance, both cannonballs launched from the same conditions (angle and velocity) would have the same range and path. Thus, in a vacuum, you cannot distinguish between the lead and wooden cannonballs based on trajectory and range alone, emphasizing how significantly air resistance influences motion on Earth.