P02-010 – Projectile Motion

Definition: projectile motion

Projectile motion refers to the description of the two-dimensional path followed by an object acted upon solely by its weight. This path is characterized by the launch angle $\theta $ and the initial velocity $\overrightarrow{v_0}$.

When a projectile, such as a golf ball, is launched with a given initial velocity $\overrightarrow{v_0}$ at a given angle $\theta $ with respect to the horizontal, it follows a parabolic path through the air that is entirely determined by $\overrightarrow{v_0}$ and $\theta $. This means that, once the golf ball is launched, there is nothing that can modify the path that it follows through the air.

Note that this require we assume that the only force being exerted on the projectile is the weight force and that any forces other than weight (e.g. propulsion, air resistance…) are neglected.

At the moment this is an unfortunate definition because it relies on the notion of force which we haven’t formally introduced. Once we do, the above definition will make more sense but for now let us agree that projectile motion implies that the projectile only experiences the effect of gravity throughout its flight.

Consequences:

1. Since gravity only acts in the vertical direction, the projectile moves at a constant speed horizontally. We then typically say that horizontally the motion of the projectile is uniform and write

a_x=0\ \ \ m/s^2

2. Along the vertical direction, the projectile is said to undergo free-fall since its acceleration is equal to the acceleration due to gravity. We then write

a_y=-g

where $g\approx -9.81\ m/s^2$ is the acceleration due to gravity (as defined in the previous chapter).

This causes the vertical velocity to decrease over time until it reaches zero at the maximum height. It then continues to decreases — which really means becoming more and more negative — until it hits the ground.

3. Overall, the motion is parabolic and the vertical position $y$ varies as a second order polynomial of $x$.
4. The initial velocity $\overrightarrow{v_0}$ has an $x$-component $v_{0x}$ and a $y$-component $v_{0y}$ that are equal to

v_{0x}=v_0{\mathrm{cos} \left(\theta \right)\ }\ \ \ \ \ \ \ \ \ \ and\ \ \ \ \ \ \ \ \ \ v_{0y}=v_0{\mathrm{sin} \left(\theta \right)\ }

as shown in the figure below