Projectile Motion

One Ball with Two Motions at Once

Projectile motion is motion along a curved path because an object moves horizontally while also moving up and down vertically. In this lesson, air resistance is ignored, so the only acceleration is gravity downward.

The easiest way to read the motion is to split it into two directions. The horizontal direction follows uniform linear motion because its velocity stays constant. The vertical direction follows uniformly accelerated motion because gravity keeps changing the vertical velocity.

x=v_0\cos\theta\ t

Here, $v_0$ is the initial speed, $\theta$ is the launch angle from the horizontal, $g$ is gravitational acceleration, $x$ is horizontal position, and $y$ is height.

The visual below uses one ball so the launch angle, peak, and range are easier to compare.

Projectile Ball Path

Change the launch angle. The ball keeps moving horizontally while gravity changes its height.

Initial speed: $v_0=22\text{ m/s}$
Time in the air: $T=3.1\text{ s}$
Range: $R=48.4\text{ m}$
Peak height: $h_{\max}=12.1\text{ m}$

Gravity Changes Only the Vertical Part

Imagine a ball leaving a ramp at an angle. Its horizontal part keeps moving forward at the same velocity, while its vertical part slows down on the way up and speeds up on the way down.

In projectile motion without air resistance, $v_x$ stays constant, but $v_y$ changes every moment.

This sentence prevents a common mistake. The object does not lose its horizontal motion at the top. Only the vertical velocity becomes zero there.

The Peak Still Moves

At the top of the path, $v_y=0$ because the ball stops for an instant in its up-and-down motion. However, $v_x$ is still present, so the ball still moves forward.

That is why the path does not stop at the top. After the top, gravity makes $v_y$ negative, which means the ball starts moving downward. At the same time, $v_x$ still carries the ball forward.

Part of the motion	What happens	Motion type
Horizontal	Velocity stays constant	Uniform linear motion
Vertical while rising	Vertical velocity gets smaller	Slowed uniformly accelerated motion
Vertical while falling	Vertical velocity grows downward	Sped-up uniformly accelerated motion

Range Comes from Time in the Air

Horizontal range, usually written as $R$ , is the horizontal distance from launch point to landing point. Because the horizontal motion has constant velocity, range is horizontal velocity multiplied by time in the air.

R=v_x t_{\text{air}}

If two balls have the same $v_x$ , the one that stays in the air longer lands farther away. If the air time is the same, the ball with the larger $v_x$ has the larger range.

A Ball Leaves a Table Horizontally

Suppose a ball leaves the edge of a table horizontally with $v_x=4\text{ m/s}$ and the table height is $1.25\text{ m}$ . With $g=10\text{ m/s}^2$ , the fall time comes from the vertical motion:

\begin{aligned} y&=\frac{1}{2}gt^2\\ 1.25&=\frac{1}{2}(10)t^2\\ t^2&=0.25\\ t&=0.5\text{ s} \end{aligned}

The horizontal range is:

R=v_x t=4(0.5)=2\text{ m}

So the ball lands $2\text{ m}$ from the foot of the table. Notice that fall time is determined by the vertical motion, while horizontal distance is determined by the horizontal motion during that time.