Projectile Motion Calculator

Give it a launch speed, an angle and a starting height, and it returns how far the projectile travels, how long it stays up, how high it gets, how fast it is moving vertically and how hard it lands. The graph redraws on every keystroke, and a time slider walks the object along its own arc so you can read off position, vertical velocity and speed at any instant. Gravity is editable for problems set on the Moon or Mars. Every equation it uses is written out below, along with a vertical velocity reference for the component most homework questions ask about first. If you have a specific textbook question, the common questions section matches the usual wordings to the formula that answers each one.

Initial speed (m/s)

Launch angle (degrees)

Launch height (m)

Gravity (m/s²)

−90°0°90°

Horizontal range

40.79 m

Peaks at 10.2 m and stays in the air 2.88 seconds.

Time of flight: 2.88 s
Max height: 10.2 m
Time to apex: 1.44 s
Horizontal velocity: 14.14 m/s
Vertical velocity at launch: 14.14 m/s
Distance at apex: 20.39 m
Impact speed: 20 m/s
Impact angle: 45° below level
Max-range angle: 45°

Trajectory

Compare with the max-range angle

Position at time

t = 0 s · x = 0 m · y = 0 m · vy = 14.14 m/s · speed = 20 m/s

The gold point marks the apex. The two axes are scaled independently to fill the box, so read the numbers rather than the shape. Air resistance is not modeled.

Projectile motion equations

Everything follows from splitting the launch velocity in two and treating the directions separately. Horizontal speed never changes; vertical speed loses g every second.

vx = v₀ cos θ vy = v₀ sin θ
x(t) = vx · t y(t) = y₀ + vy · t − ½ g t²
time to apex = vy ÷ g
max height = y₀ + vy² ÷ 2g
time of flight = (vy + √(vy² + 2 g y₀)) ÷ g
range = vx × time of flight
impact speed = √(v₀² + 2 g y₀)

With y₀ = 0 the last three collapse to the versions in most textbooks: t = 2v₀ sin θ ÷ g, R = v₀² sin(2θ) ÷ g, and the projectile lands at the speed it was launched at.

The one projectile motion equation to memorize is the vertical position line, y(t) = y₀ + v₀ sin θ · t − ½ g t². Setting it to zero gives the flight time, differentiating it gives the vertical velocity, and setting that to zero gives the apex. The rest is bookkeeping.

Symbol	Meaning	Units
v₀	Initial speed, the launch velocity	m/s or ft/s
θ	Launch angle measured from the horizontal	degrees
y₀	Launch height above the landing point	m or ft
g	Gravitational acceleration, 9.80665 on Earth	m/s² or ft/s²
vx, vy	Horizontal and vertical velocity components	m/s or ft/s

Common projectile questions, solved

Most homework problems are one of these six, each answered by a single formula. They use g = 9.8 m/s² because that is what homework sets use. The calculator defaults to the standard 9.80665, so type 9.8 into its gravity field when you want to match a textbook answer to the last digit.

How long is a projectile in the air?

Given the initial vertical velocity

t = 2 · vy ÷ g

Launched with vy = 24.5 m/s: t = 2 × 24.5 ÷ 9.8 = 5 s. This holds when it lands at the height it left from. From a ledge or cliff, use the full time of flight equation above.

What was the horizontal velocity?

Given how far away it landed and the time in the air

vx = d ÷ t

A ball lands 30 m away after 2.5 s: vx = 30 ÷ 2.5 = 12 m/s. Gravity never touches the horizontal component, so g doesn't appear.

How far did it travel horizontally?

Given the horizontal velocity and the time in the air

d = vx × t

Moving at 35 m/s horizontally for 2.2 s: d = 35 × 2.2 = 77 m. If the question also gives a vertical velocity, ignore it. It only matters when you have to find the time yourself.

What was the initial vertical velocity?

Given the maximum height

vy = √(2 · g · h)

It peaks 5 m up: vy = √(2 × 9.8 × 5) = 9.9 m/s. Vertical velocity is zero at the top, so this is the max height equation run backward.

What is the angle of impact?

Launched horizontally from a height

θ = arctan(√(2 · g · h) ÷ vx)

Rolled off a 5 m ledge at 6 m/s: it hits with a vertical speed of √(2 × 9.8 × 5) = 9.9 m/s, so θ = arctan(9.9 ÷ 6) = 58.8° below the horizontal. A question that expects a negative angle wants −58.8°.

What is the vertical velocity after t seconds?

Dropped, or launched horizontally

vy = −g · t

Cargo released from a plane flying at 200 m/s, 4 s later: vy = −9.8 × 4 = −39.2 m/s. The plane's speed doesn't change the answer, because horizontal motion has no effect on the vertical.

A worked example, start to finish

A ball is kicked from level ground at 20 m/s, 30° above the horizontal. Take g as 9.807 m/s².

Step 1. Split the velocity

vx = 20 cos 30° = 17.32 m/s vy = 20 sin 30° = 10 m/s
Step 2. Time to the top

t = vy ÷ g = 10 ÷ 9.807 = 1.02 s
Step 3. Maximum height

h = vy² ÷ 2g = 100 ÷ 19.613 = 5.1 m
Step 4. Total flight time

T = 2 × 1.02 = 2.04 s
Step 5. Range

R = vx × T = 17.32 × 2.04 = 35.32 m

The one-line check: R = v₀² sin(2θ) ÷ g = 400 × sin 60° ÷ 9.807 = 35.32 m. That shortcut only holds when the projectile lands at the height it left from, which is why the calculator solves the quadratic instead.

Vertical velocity, launch to landing

Vertical velocity is the part of the motion gravity actually touches, so it is what most questions turn on. It starts at v₀ sin θ, loses g every second, passes through zero at the apex and keeps going negative on the way down. Three equations cover every version of the question:

at launch: vy = v₀ sin θ
at time t: vy(t) = v₀ sin θ − g t
at landing: vy = −√(v₀² sin²θ + 2 g y₀)

Positive is upward. Used as a vertical velocity calculator, the tool reports the launch value in the results panel and reads out vy at whatever instant you drag the time slider to. Here is the whole flight of a 20 m/s shot at 45° from level ground:

Time (s)	Vertical velocity (m/s)	Height (m)	Speed (m/s)
0	14.14	0	20
0.5	9.24	5.85	16.89
1	4.34	9.24	14.79
1.44	0	10.2	14.14
2	-5.47	8.67	15.16
2.88	-14.14	0	20

Two things to notice. Vertical velocity is zero at 1.44 s, which is exactly where the height peaks: that is the definition of the apex, not a coincidence. And the speed at landing matches the speed at launch, because from level ground the flight is symmetric. The horizontal component sat at 14.14 m/s the entire time, untouched. The speed column combines the two components the way any right triangle does, speed = √(vx² + vy²), so it is the Pythagorean theorem applied to a velocity triangle rather than a distance one.

The same throw at five different angles

All five rows launch at 20 m/s from level ground. Notice that 15° and 75° reach the same distance, as do 30° and 60°: from level ground, complementary angles always do. Every row is generated at build time by the calculator's solver.

Angle	Range (m)	Max height (m)	Time of flight (s)
15°	20.39	1.37	1.06
30°	35.32	5.1	2.04
45°	40.79	10.2	2.88
60°	35.32	15.3	3.53
75°	20.39	19.03	3.94

Why 45° stops being the best angle

The 45° rule is only true when the projectile finishes at the height it started. Throw from a ledge, a cliff or shoulder height and part of the flight is a free fall the launch never had to pay for, so a flatter angle wins: it converts that borrowed time into forward distance. The angle that maximizes range is arctan(v₀ ÷ √(v₀² + 2 g y₀)), and the calculator reports it for whatever you enter.

Launch height	Best angle	Range at that angle	Range at 45°
0 m	45°	40.79 m	40.79 m
2 m	43.7°	42.74 m	42.7 m
10 m	39.3°	49.79 m	49.1 m
50 m	28.3°	75.78 m	69.95 m

At 20 m/s the gain is modest, a meter or two, but the angle keeps sliding downward as the launch point rises. It is the reason a shot putter releases nearer 42° than 45°: the shot leaves the hand roughly two meters above where it lands.

Reading the trajectory graph

The solid teal curve is your shot. The gold point on it is the apex, with the dashed vertical line marking how far downrange the peak happens.
The dashed gray curve is the same speed and height launched at the max-range angle, so you can see what a different angle costs or buys. Uncheck the box to hide it.
Drag the time slider to move the object along the path. The line underneath reads out position and speed at that instant, which is what most homework questions ask for.
The axes are scaled independently so the arc fills the box. A 200 m by 20 m flight would be a flat sliver at true scale, so read the tick values rather than judging steepness by eye.

How to use this calculator

Pick meters or feet. The gravity field re-anchors itself to that system; your speed and height stay as you typed them.
Enter the initial speed and the launch angle, or drag the angle slider to sweep it from −90° to 90°.
Set the launch height. Leave it at 0 for a ground-level problem, or enter the ledge, table or cliff height.
Read the range, flight time, apex and impact figures, then scrub the time slider for any moment in between.

What this model leaves out

Gravity is the only force here, which is the model behind every introductory physics course and almost every homework problem. Air resistance, spin, wind and the curvature of the Earth are not included. Drag would shorten the range, pull the peak past the halfway mark and make the descent steeper than the climb, and modeling it takes a mass, a drag coefficient and a step-by-step numerical solution rather than a formula. For dense objects over short distances the no-drag answer is close. For a golf ball, an arrow or anything light and fast, treat it as an upper bound.

How this calculator is checked

The worked example and both tables on this page are generated at build time by the same solver the calculator runs, so the numbers here match what the tool gives you.
The formulas are unit-tested against the closed-form textbook results, including the range and height identities, complementary-angle pairs and the energy check on impact speed.
The six common-question answers come from the same solver with g set to 9.8, the value homework sets use, so they match an answer key rather than drifting in the last digit.
Edge cases are covered on purpose: zero speed, a downward launch, a horizontal launch from a height, and a gravity of zero, which is rejected rather than dividing by zero.
Everything runs in your browser. Nothing you enter is stored or sent to a server.

Last reviewed: October 2026. Found an error? Report it and we'll fix it.

Frequently Asked Questions

What is projectile motion?

Projectile motion is the path an object follows once it has been launched and only gravity acts on it. The horizontal velocity stays constant and the vertical velocity changes at a steady rate, which combine into a parabola. Splitting the launch velocity into those two independent parts is the whole method.

What is the formula for the range of a projectile?

From ground level the range is R = v₀² sin(2θ) ÷ g. At 20 m/s and 30 degrees that gives R = 400 × sin 60° ÷ 9.80665 = 35.32 m. When the launch point is above the landing point the shortcut no longer applies, so the calculator solves the vertical position for zero and multiplies the resulting flight time by the horizontal velocity.

How do you find the time of flight?

Solve y₀ + v₀ sin(θ) t − ½ g t² = 0 for the positive root, which gives t = (v₀ sin θ + √(v₀² sin²θ + 2 g y₀)) ÷ g. From the ground this simplifies to twice the time to the top, t = 2 v₀ sin θ ÷ g.

How high does a projectile go?

The peak is y₀ + v₀² sin²θ ÷ 2g, reached at t = v₀ sin θ ÷ g when the vertical velocity passes through zero. Horizontal speed has no effect on the height, which is why a bullet fired level and a bullet dropped hit the ground at the same moment.

How do you calculate vertical velocity?

Vertical velocity at launch is v₀ sin θ, and at any later moment it is v₀ sin θ − g t, taking upward as positive. At 20 m/s and 45 degrees that starts at 14.14 m/s and drops by 9.80665 m/s every second. The results panel shows the launch value and the time slider under the graph reads out vy at whatever instant you drag it to.

What is the vertical velocity at the highest point?

Zero. That is what defines the apex: the projectile has stopped rising and has not started falling. Its speed is not zero there though, because the horizontal component is untouched by gravity and carries on at v₀ cos θ.

Why doesn't the horizontal velocity change?

Because with no air resistance nothing pushes horizontally. Gravity acts straight down, so it only changes the vertical component. That independence is what lets you solve the vertical motion for time and then multiply by a constant horizontal speed to get the range.

Do I have to enter an angle?

No. Set the angle to 0 for a horizontal launch, which is the classic ball-rolls-off-a-table problem, or set the speed to 0 to use it as a plain projectile calculator for an object dropped from a height. Negative angles down to −90 degrees cover anything thrown downward.

Is 45 degrees always the best angle?

Only when the projectile starts and lands at the same height. Launch from higher up and the best angle drops below 45 degrees, because a flatter shot spends its extra hang time moving forward. Throwing at 20 m/s from 10 m, 39.3 degrees reaches 49.79 m against 49.1 m at 45 degrees. The calculator marks that angle and draws it as the dashed path.

Why do two different angles give the same range?

From level ground, complementary angles are a matched pair: 30 and 60 both give 35.32 m at 20 m/s, because sin(2θ) has the same value for both. The steeper one takes longer and flies higher, so it is the flatter one that gets there first.

How fast is it going when it lands?

Impact speed is √(v₀² + 2 g y₀), which depends on the launch height but not on the angle. Launched from the ground it lands at exactly the speed it left at, mirrored below the horizontal. The calculator also reports the angle of descent.

Does this account for air resistance?

No. This is the standard no-drag model used in physics classes and homework, where gravity is the only force. Real drag shortens the range and makes the descent steeper than the climb, and it depends on the object's mass, shape and speed. For a ball, a shot put or a thrown object over short distances the difference is small. For a golf ball or an arrow it is not.

Can I use this for other planets?

Yes. The gravity field is editable, so enter 1.62 for the Moon, 3.72 for Mars or 24.79 for Jupiter in meters per second squared and every result updates. The default is standard Earth gravity, 9.80665 m/s² or 32.174 ft/s².

Is anything I enter sent to a server?

No. The solver and the graph both run in your browser. Nothing you type leaves the page.