Projectile Motion

Projectile motion is the motion of an object launched into the air with some initial

velocity and then left to move freely under gravity alone, air resistance being neglected.

It is the most important application of two-dimensional motion with constant acceleration

in this chapter, and it is solved directly using the principle of independence of motion

(Section 3.11).

Setup. A projectile is launched from the ground with initial speed $u$ at an angle

$\theta$ above the horizontal (see the figure). Its initial velocity resolves into a

horizontal component $u_x = u\cos\theta$ and a vertical component $u_y = u\sin\theta$. The

only acceleration acting is gravity, directed vertically downward: $a_x = 0$ (no horizontal

acceleration) and $a_y = -g$ (taking "up" as positive).

Horizontal motion therefore proceeds at constant velocity,

while vertical motion is uniformly accelerated, exactly like a ball thrown straight up:

Trajectory. Eliminating $t$ between $x(t)$ and $y(t)$ gives the equation of the path:

which is the equation of a parabola — confirming that every projectile (launched at any angle

other than exactly vertical) traces a parabolic arc.

Time of flight $T$ is the total time the projectile stays in the air, found by setting

$y = 0$ (landing at the same height as launch):

Maximum height $H$ is reached when $v_y = 0$ (the topmost point of the path), where the

velocity is purely horizontal, $u\cos\theta$:

Horizontal range $R$ is the horizontal distance covered in the full time of flight,

$R = (u\cos\theta)\,T$, which simplifies to

Because $\sin 2\theta$ reaches its maximum value of 1 when $2\theta = 90^\circ$, the range is

greatest at a launch angle of $45^\circ$, for a fixed launch speed $u$. The table shows …