# Unit 7 - Projectile Motion

Projectile Motion Concepts

Author: Marc Reif

Paradigm: The paradigm is the video analysis of a projectile. This allows us to separate the horizontal (X) and vertical (Y) components of the motion.

X-t and Vx- t graphs Y-t and Vy-t graphs

A projectile is launched with some initial velocity, but is significantly affected only by gravity and air resistance after launch. For most of our work we will consider air resistance negligible.

Objects such as Frisbees, which are designed to have high air resistance in order to "foil" gravity, are not considered projectiles.

Projectiles may be launched at an angle to the horizontal. The initial velocity is the "vector sum" (resultant) of the horizontal and vertical components of the velocity. See the diagram and equations below:

The path of a projectile is a parabola, or a portion of a parabola. Galileo, the founder of modern science, was the first to put this into print.

If air resistance is negligible, gravity is the only force acting on the projectile.

If the only force is gravity, which acts in the vertical direction, there is nothing to change the horizontal velocity of the projectile while it is in motion. The horizontal velocity is constant throughout the motion.

In the horizontal direction (X), the projectile is described by the Constant Velocity Model. Anything that is described by the CV model is explained by the Free Particle model.

The vertical velocity is affected by the force of earth's gravity on the projectile. Because the force of gravity is constant, in the vertical direction (Y), the projectile is described by the Constant Acceleration Model. The value of the acceleration is g, the freefall acceleration for the planet you are on (approx. -9.8 m/s2 on earth). Anything that is described by the CA model is explained by the Constant Force model.

The speed of a projectile is at a minimum at the highest point of its path. This is because the vertical velocity is at a minimum.

If a projectile is launched upwards at any angle greater than 0 degrees from the horizontal, its vertical velocity is zero for an instant at the top of its path, as it changes from an upward direction to a downward direction.

If a projectile is launched upwards at any angle greater than 0 degrees from the horizontal, at any height (y position), the magnitude of the upward velocity is equal to the magnitude of the downward velocity at the same height.

We often call the launch height zero for a projectile that returns to the same level it was launched from. This simplifies equations.

In the absence of air resistance, launching a projectile with an angle of 45 degrees gives the greatest range. Can you explain why?

Complementary angles (30 and 60, 27 and 63, 10 and 80) give the same range, but time of flight differs. Again, can you explain why?