Projectile Motion

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In a previous atom we discussed what the various components of an object in projectile motion are. In this atom we will discuss the basic equations that go along with them in the special case in which the projectile initial positions are null (i.e. $x_0 = 0$ and $y_0 = 0$ ) .

Initial Velocity

The initial velocity can be expressed as x components and y components:

$u_x = u \cdot \cos\theta \\ u_y = u \cdot \sin\theta$

In this equation, $u$ stands for initial velocity magnitude and $\small{\theta}$ refers to projectile angle.

Time of Flight

The time of flight of a projectile motion is the time from when the object is projected to the time it reaches the surface. As we discussed previously, $T$ depends on the initial velocity magnitude and the angle of the projectile:

$\displaystyle {T=\frac{2 \cdot u_y}{g}\\ T=\frac{2 \cdot u \cdot \sin\theta}{g}}$

Acceleration

In projectile motion, there is no acceleration in the horizontal direction. The acceleration, $a$, in the vertical direction is just due to gravity, also known as free fall:

 $\displaystyle {a_x = 0 \\ a_y = -g}$

Velocity

The horizontal velocity remains constant, but the vertical velocity varies linearly, because the acceleration is constant. At any time, $t$, the velocity is:

$\displaystyle {u_x = u \cdot \cos{\theta} \\ u_y = u \cdot \sin {\theta} - g \cdot t}$

You can also use the Pythagorean Theorem to find velocity:

$u=\sqrt{u_x^2+u_y^2}$

Displacement

At time, t, the displacement components are:

$\displaystyle {x=u \cdot t \cdot \cos\theta\\ y=u \cdot t \cdot \sin\theta-\frac12gt^2}$

The equation for the magnitude of the displacement is $\Delta r=\sqrt{x^2+y^2}$ .

Parabolic Trajectory

We can use the displacement equations in the x and y direction to obtain an equation for the parabolic form of a projectile motion:

$\displaystyle y=\tan\theta \cdot x-\frac{g}{2 \cdot u^2 \cdot \cos^2\theta} \cdot x^2$

Maximum Height

The maximum height is reached when $v_y=0$ . Using this we can rearrange the velocity equation to find the time it will take for the object to reach maximum height

$\displaystyle t_h=\frac{u \cdot \sin\theta}{g}$

where $t_h$ stands for the time it takes to reach maximum height. From the displacement equation we can find the maximum height

$\displaystyle h=\frac{u^2 \cdot \sin^2\theta}{2\cdot g}$ 

Range

The range of the motion is fixed by the condition $\small{\sf{y = 0}}$ . Using this we can rearrange the parabolic motion equation to find the range of the motion:

$\displaystyle R=\frac{u^2 \cdot \sin2\theta}{g}$ .

Range of Trajectory

The range of a trajectory is shown in this figure.