Horizontal Projectiles

The video lesson for this topic is coming soon.

Essential Question

What path does an object take when it’s launched sideways?

A ball rolling off a table moves forward and falls at the same time. By the end of this lesson, you should be able to treat the horizontal and vertical motions separately and solve for time in the air, horizontal distance, and vertical drop.

What Is a Projectile?

Any object flying through the air
Only gravity acts on it
Its path is a curve: a parabola

A ball rolls off the edge of a table and follows a dashed curved path down to the floor. The curve is labeled parabola. Caption: gravity bends the path into a parabola.

A projectile is any object moving through the air with only gravity acting on it, like a thrown ball, a kicked football, or a ball rolling off a table. As long as we ignore air resistance, the path of every projectile is the same kind of curve, called a parabola. In this lesson the projectile starts out moving horizontally.

Dropped vs. Launched

Both start with v_y = 0
Both fall at the same rate
They hit the ground together

A strobe picture of two balls released at the same moment. A red ball is dropped straight down and a blue ball is launched sideways along a parabola. Dashed horizontal lines show both balls at the same height at each moment, and they land at the same time. Caption: same height at every moment.

Drop one ball and launch another sideways from the same height at the same moment. The launched ball moves forward, but at every instant both balls are at exactly the same height, and they hit the ground at the same time. Neither ball starts with any vertical velocity, and gravity pulls on both the same way. Moving sideways does not change how fast an object falls.

Horizontal Motion

No horizontal force
v_x stays constant
Equal sideways steps

The parabolic path of a horizontal projectile with five evenly timed ball positions. Each ball has a blue horizontal arrow of the same length labeled v x. Below, tick marks show the balls are equally spaced sideways. Caption: v x never changes.

Once the projectile is launched, nothing pushes or pulls it sideways (we are ignoring air resistance). With no horizontal force, there is no horizontal acceleration, so the horizontal velocity v_x stays exactly the same the whole flight. That is why the ball moves the same distance sideways in each equal time interval. Horizontally, a projectile is just constant velocity: Δx = v_xt.

Vertical Motion

Gravity pulls straight down
v_y grows 9.8 m/s each second
Just like a dropped object

The same parabolic path with five evenly timed ball positions. Each ball has a red downward arrow labeled v y. The first ball has no red arrow, and the arrows get longer at each later position. Caption: v y grows, just like free fall.

Vertically, the projectile is in free fall. It starts with v_y = 0, because it was launched horizontally, and gravity makes its downward velocity grow by 9.8 m/s every second. The red arrows get longer and longer, and the ball drops farther in each time step. Vertically, a horizontal projectile behaves exactly like an object dropped from rest: a = −9.8 m/s² and Δy = ½at².

Putting It Together

The two motions are independent
v_x: same; v_y: grows
Total velocity is tangent to the path

The parabolic path with five ball positions. Each ball has a blue horizontal arrow that stays the same length and a red downward arrow that grows. At the fourth position, a purple arrow labeled v goes from the ball to the tip of the combined components and runs along the curve.

The horizontal and vertical motions are independent: neither one affects the other. At every point, the velocity has a constant blue horizontal component and a growing red vertical component. Adding them tip to tail gives the total velocity (purple), which always points along the path, tangent to the curve. Because the vertical part keeps growing while the horizontal part stays the same, the path bends more and more steeply downward.

Solving Horizontal Projectiles

Solve the horizontal and vertical parts separately, then connect them with time.

Horizontal: v_x is constant, a_x = 0, so Δx = v_xt
Vertical: v_iy = 0, a_y = −9.8 m/s², so Δy = ½a_yt²
Time: t is the same in both equations

Two cards side by side. The horizontal card says v x is constant, a x equals 0, and delta x equals v x t. The vertical card says v i y equals 0, a y equals −9.8 m/s squared, and delta y equals one half a y t squared. A yellow box below says time t is the same for both.

Usually the height tells you the time (from the vertical equation), and then the time tells you how far the projectile goes sideways (from the horizontal equation). Remember that Δy is negative because the projectile ends up below its launch point.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Marble off a Table

A marble rolls off a lab table 0.784 m high at 1.50 m/s. How long is the marble in the air?

A marble rolls off the edge of a table at 1.50 m/s and follows a dashed parabola to the floor. A double arrow beside the path shows the table height, 0.784 m.

Knowns (vertical): v_iy = 0 m/s, Δy = −0.784 m, a_y = −9.8 m/s², t = ?
Equation: Δy = ½a_yt², rearranged to t = √(2Δy / a_y)
Substitute: t = √(2(−0.784 m) / (−9.8 m/s²)) = √(0.160 s²)
Answer: t = 0.400 s. The 1.50 m/s does not affect the time in the air.

Example 2: Supply Drop

A plane flying horizontally at 60.0 m/s releases a supply package from a height of 490 m. How far does the package travel horizontally before it lands?

A plane moving right at 60.0 m/s drops a package that follows a dashed parabola to the ground. A vertical double arrow shows the height, 490 m, and a horizontal double arrow along the ground is labeled delta x = ?.

Vertical knowns: v_iy = 0 m/s, Δy = −490 m, a_y = −9.8 m/s²
Find time: t = √(2Δy / a_y) = √(2(−490 m) / (−9.8 m/s²)) = √(100 s²) = 10.0 s
Horizontal knowns: v_x = 60.0 m/s (the package keeps the plane’s speed), t = 10.0 s
Find Δx: Δx = v_xt = (60.0 m/s)(10.0 s)
Answer: Δx = 600. m

From the plane, the package looks like it falls straight down, because it keeps moving forward at the plane’s speed the whole time.

Example 3: Dart Drop

A dart is thrown horizontally at 15.0 m/s toward a dartboard 3.00 m away. How far does the dart drop by the time it reaches the board?

A dart leaves the thrower's hand moving right at 15.0 m/s. A straight dashed line aims at the bullseye, but the dart's dashed parabola curves down and hits the board lower. The gap is labeled drop = ?, and the distance to the board is 3.00 m.

Horizontal knowns: v_x = 15.0 m/s, Δx = 3.00 m
Find time: t = Δx / v_x = 3.00 m / 15.0 m/s = 0.200 s
Vertical knowns: v_iy = 0 m/s, a_y = −9.8 m/s², t = 0.200 s
Find Δy: Δy = ½a_yt² = ½(−9.8 m/s²)(0.200 s)² = (−4.9 m/s²)(0.0400 s²) = −0.196 m
Answer: The dart drops 0.196 m (about 20 cm) below where it was aimed.

Example 4: Launch Speed

A ball is kicked horizontally off a ledge 4.9 m high. It lands 8.0 m from the base of the ledge. How fast was it kicked?

A ball at the top edge of a 4.9 m tall ledge with an arrow labeled v = ? pointing right. A dashed parabola goes from the ball to the ground, landing 8.0 m from the base of the ledge.

Vertical knowns: v_iy = 0 m/s, Δy = −4.9 m, a_y = −9.8 m/s²
Find time: t = √(2Δy / a_y) = √(2(−4.9 m) / (−9.8 m/s²)) = √(1.0 s²) = 1.0 s
Horizontal knowns: Δx = 8.0 m, t = 1.0 s, v_x = ?
Find v_x: v_x = Δx / t = 8.0 m / 1.0 s
Answer: The ball was kicked at 8.0 m/s.

Big Ideas

Horizontal and vertical motions are independent
Horizontal: v_x constant, Δx = v_xt
Vertical: free fall, Δy = ½a_yt²
Time links the two motions

Back to the essential question: an object launched sideways follows a curved path called a parabola. Its horizontal velocity stays constant because no horizontal force acts on it, while its vertical motion is ordinary free fall with a = −9.8 m/s². The time in the air links the two motions. Next up, we’ll launch projectiles at an angle.