Projectile Motion

Launch angles, ranges and heights: explore every arc a ball can trace.

18 minUpdated 21 October 2026time of flightrangemaximum heighttrajectory

Lesson at a glanceShow the key points

Horizontal velocity u cos θ stays constant; vertical motion is free fall starting at u sin θ.
Time of flight T = 2u sin θ / g; time to the top is T/2.
Range R = u² sin 2θ / g, largest (u²/g) at θ = 45°; angles θ and 90° − θ give the same range.
Maximum height H = u² sin²θ / 2g.
The path is a parabola: y = x tan θ − g x² / (2u² cos²θ).

A cricket ball, a water fountain, a long jumper: anything launched at an angle traces the same family of curves. They are all one horizontal motion and one vertical motion, running side by side.

What makes a projectile

Throw something at an angle near the Earth’s surface and it follows a curved path. Such an object is a projectile. We will assume it stays close to the ground and that air resistance is negligible. Then its acceleration is constant: $g$ , straight down. So projectile motion is just motion in a plane with constant acceleration, exactly what the previous lesson handled.

Definition · The vocabulary

The launch point O is the point of projection; the angle θ above the horizontal is the angle of projection; the horizontal distance OB to where it lands at the same level is the horizontal range R; the total time in the air is the time of flight T; and the peak A is at the maximum height H.

Interactive

Projectile launcher

Pick a speed, an angle and a world, then launch. Previous shots leave faint trails so you can compare. Watch the arrows: the horizontal velocity (blue) never changes, while the vertical one (coral) shrinks, flips and grows.

time of flight T2.04s

range R35.35m

max height H5.10m

v_x

17.32m/s

v_y

10.00m/s

Launch speed

u

20.0m/s

Launch angle

\theta

30°

World

Splitting the motion

Put the origin at the launch point, the x-axis horizontal and the y-axis vertically up, and start the clock at launch. With launch speed $u$ at angle $\theta$ :

a_x = 0,\quad a_y = -g, \qquad u_x = u\cos\theta,\quad u_y = u\sin\theta

Horizontal: steady

With no horizontal acceleration, the horizontal velocity never changes:

v_x = u\cos\theta, \qquad x = (u\cos\theta)\,t

Vertical: thrown straight up

Vertically it is exactly a ball thrown straight up with speed $u\sin\theta$ :

v_y = u\sin\theta - gt, \qquad y = (u\sin\theta)\,t - \tfrac12gt^2, \qquad v_y^2 = u^2\sin^2\theta - 2gy

Interactive

A projectile and its two shadows

Light from straight above casts a shadow on the ground (blue). Light from the side casts one on the back wall (coral). The ground shadow glides at constant speed; the wall shadow rises and falls like a ball thrown straight up. Drag to orbit.

Loading 3D scene…

Speed

u

12.0m/s

Angle

\theta

55°

Three results worth knowing

Time of flight

It lands when $y = 0$ again: $0 = t\,(u\sin\theta - \tfrac12gt)$ . One root, $t = 0$ , is the launch. The other is the landing:

Time of flight(3.22)

T = \frac{2u\sin\theta}{g}

Range

Horizontal distance covered in that time, using $2\sin\theta\cos\theta = \sin 2\theta$ :

Horizontal range(3.23)

R = (u\cos\theta)\,T = \frac{u^2\sin 2\theta}{g}

Maximum height

At the top the velocity is momentarily horizontal, so $v_y = 0$ , which happens at $t = u\sin\theta/g$ (3.24), exactly half the time of flight. The climb and the fall take equal times. Substituting:

Maximum height(3.25)

H = \frac{u^2\sin^2\theta}{2g}

Interactive

Which angle throws farthest?

Range (green) peaks at 45°, and angles that add up to 90° land at exactly the same spot. Height (gold) just keeps rising with angle.

R(30°)35.35m

R(60°)35.35m

R max (45°)40.82m

Speed

u

20.0m/s

Angle

\theta

30°

Worked example

A kick at 30°

A football is kicked at 20 m/s, 30° above the horizontal. Find T, R and H. Take g = 10 m/s².

What to remember

1Horizontal motion: constant velocity u cos θ. Vertical motion: free fall starting at u sin θ.
2 $T = 2u\sin\theta/g,\ \ R = u^2\sin2\theta/g,\ \ H = u^2\sin^2\theta/2g$ .
3Maximum range at 45°; complementary angles share a range; the path is a parabola.

Check yourself

At the highest point of a projectile’s path, its velocity is…

The same ball is thrown at the same speed and angle on the Moon (g is about 1/6 of Earth’s). Its range is…

Two balls are launched at 20° and 70° with the same speed. Which lands farther?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchProjectile at an angleKhan Academy PlayProjectile MotionPhET · CU Boulder ReadProjectile motionOpenStax University Physics §4.3 ReadCharacteristics of a projectile’s trajectoryThe Physics Classroom ReadTrajectoriesHyperPhysics ReadProjectile motionWikipedia

PreviousMotion in a Plane Next Changing Frames

Projectile Motion

Launch angles, ranges and heights: explore every arc a ball can trace.

18 minUpdated 21 October 2026time of flightrangemaximum heighttrajectory

Lesson at a glanceShow the key points

Horizontal velocity u cos θ stays constant; vertical motion is free fall starting at u sin θ.
Time of flight T = 2u sin θ / g; time to the top is T/2.
Range R = u² sin 2θ / g, largest (u²/g) at θ = 45°; angles θ and 90° − θ give the same range.
Maximum height H = u² sin²θ / 2g.
The path is a parabola: y = x tan θ − g x² / (2u² cos²θ).

A cricket ball, a water fountain, a long jumper: anything launched at an angle traces the same family of curves. They are all one horizontal motion and one vertical motion, running side by side.

What makes a projectile

Definition · The vocabulary

Interactive

Projectile launcher

time of flight T2.04s

range R35.35m

max height H5.10m

v_x

17.32m/s

v_y

10.00m/s

Launch speed

u

20.0m/s

Launch angle

\theta

30°

World

Splitting the motion

Put the origin at the launch point, the x-axis horizontal and the y-axis vertically up, and start the clock at launch. With launch speed $u$ at angle $\theta$ :

a_x = 0,\quad a_y = -g, \qquad u_x = u\cos\theta,\quad u_y = u\sin\theta

Horizontal: steady

With no horizontal acceleration, the horizontal velocity never changes:

v_x = u\cos\theta, \qquad x = (u\cos\theta)\,t

Vertical: thrown straight up

Vertically it is exactly a ball thrown straight up with speed $u\sin\theta$ :

v_y = u\sin\theta - gt, \qquad y = (u\sin\theta)\,t - \tfrac12gt^2, \qquad v_y^2 = u^2\sin^2\theta - 2gy

Interactive

A projectile and its two shadows

Loading 3D scene…

Speed

u

12.0m/s

Angle

\theta

55°

Three results worth knowing

Time of flight

It lands when $y = 0$ again: $0 = t\,(u\sin\theta - \tfrac12gt)$ . One root, $t = 0$ , is the launch. The other is the landing:

Time of flight(3.22)

T = \frac{2u\sin\theta}{g}

Range

Horizontal distance covered in that time, using $2\sin\theta\cos\theta = \sin 2\theta$ :

Horizontal range(3.23)

R = (u\cos\theta)\,T = \frac{u^2\sin 2\theta}{g}

Maximum height

At the top the velocity is momentarily horizontal, so $v_y = 0$ , which happens at $t = u\sin\theta/g$ (3.24), exactly half the time of flight. The climb and the fall take equal times. Substituting:

Maximum height(3.25)

H = \frac{u^2\sin^2\theta}{2g}

Interactive

Which angle throws farthest?

Range (green) peaks at 45°, and angles that add up to 90° land at exactly the same spot. Height (gold) just keeps rising with angle.

R(30°)35.35m

R(60°)35.35m

R max (45°)40.82m

Speed

u

20.0m/s

Angle

\theta

30°

Worked example

A kick at 30°

A football is kicked at 20 m/s, 30° above the horizontal. Find T, R and H. Take g = 10 m/s².

What to remember

1Horizontal motion: constant velocity u cos θ. Vertical motion: free fall starting at u sin θ.
2 $T = 2u\sin\theta/g,\ \ R = u^2\sin2\theta/g,\ \ H = u^2\sin^2\theta/2g$ .
3Maximum range at 45°; complementary angles share a range; the path is a parabola.

Check yourself

At the highest point of a projectile’s path, its velocity is…

The same ball is thrown at the same speed and angle on the Moon (g is about 1/6 of Earth’s). Its range is…

Two balls are launched at 20° and 70° with the same speed. Which lands farther?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousMotion in a Plane Next Changing Frames

What makes a projectile#

Splitting the motion#

Horizontal: steady

Vertical: thrown straight up

Three results worth knowing#

Time of flight

Range

Maximum height

Check yourself

Keep exploring

What makes a projectile#

Splitting the motion#

Horizontal: steady

Vertical: thrown straight up

Three results worth knowing#

Time of flight

Range

Maximum height

Check yourself

Keep exploring

What makes a projectile

Splitting the motion

Three results worth knowing

What makes a projectile

Splitting the motion

Three results worth knowing