7.01: Projectile Motion

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People have used projectile weapons throughout history.

Throughout history, armies have used weapons, called projectiles, that are capable of launching objects or explosives over large distances.

- The Romans used catapults and large stationary crossbows called ballistae.

- In medieval times, people used trebuchets to hurl large stones at castle walls during a siege.

- During the late middle ages and early Renaissance, people replaced trebuchets with cannons and gunpowder.

Military strategists, philosophers, and scientists asked questions about how projectiles moved.

- These questions were: what path did they take? How long did projectiles remain in the air?

- Such questions had practical applications for artillery, such as determining what angle a projectile must be launched to reach the maximum distance.

What path does a projectile take?

Ancient, medieval, and Renaissance artillery experts noticed that their projectiles followed a curved path or better known as a trajectory, but they could not determine why this was so.

Galileo was the first person to describe the physics of projectile motion.

Galileo studied the properties of motion.

Galileo studied the motions of objects extensively by rolling balls on horizontal surfaces and down inclined planes. He also dropped them from heights and found conclusions from his results. His conclusions were as follows:

1. Objects fell at the same rate regardless of the mass.

2. If an object was at rest it would remain at rest. If it was in motion at constant velocity in a straight line, it would remain in that motion unless an outside force acted on it. Galileo's findings on inertia were later restated by Isaac Newton as his first law of motion.

3. Objects undergoing constant acceleration (or free fall) could be described by the kinematic equations shown in the table.

Galileo applied these ideas to motion in 2 dimensions and came to a remarkable conclusion

Kinematic Equations for Uniform (Constant) Acceleration:

vf = vi + a∆t vf = vi+g∆t

∆d = 1/2(vf+vi)∆t ∆d = 1/2(vf+vi)∆t

∆d = vi∆t + 1/2a∆t^2 ∆d = vi∆t + 1/2g∆t^2

v2^t = v^2i + 2a∆d v^2t = v^2i + 2g∆d

Also useful: Also useful:

v¯ = vf+vi/2 v¯ = vf + vi2

Horizontal and vertical motions are independent.

Galileo realized that motion has horizontal and vertical components and that these components are independent.

If you roll a ball at constant velocity off a frictionless table, it covers equal distances in equal time intervals. Since the ball moves at constant velocity, there can be no net force acting on it in a horizontal direction.

If you drop a ball it accelerates downward, so gravity acts in a vertical direction.

As a ball rolls on a table, gravity pulls it against the table. Gravity does not slow the ball down though, because gravity is not acting in the direction of the ball's motion (ignoring friction).

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