Projectile Motion Calculator

About the Projectile Motion Calculator

A kicked football, a basketball shot, water from a hose and a stone thrown off a cliff all follow the same curved path once they leave the launcher. Ignoring air resistance, that path is a parabola, and every feature of it can be calculated from the launch speed, the launch angle and the starting height.

This calculator works through the standard method used in physics courses. It resolves the launch velocity into horizontal and vertical components, finds when the projectile reaches its highest point and when it lands, and then combines the two directions to give the range and the speed and angle at impact. You can change the value of gravity, for example to 1.62 m/s² to see how far the same throw would go on the Moon.

What is projectile motion?

A projectile is an object that moves only under gravity after it is launched. Its horizontal velocity stays constant because nothing pushes it sideways, while its vertical velocity decreases steadily at $g \approx 9.81$ m/s² downward. The two motions happen at the same time but independently, linked only by the shared clock.

Because the horizontal position grows linearly with time and the vertical position is quadratic in time, the path is a parabola. On level ground, the range is $R = \frac{v_0^2 \sin 2\theta}{g}$ , which is largest at $\theta = 45°$ , and complementary angles such as 30° and 60° give the same range with different heights and flight times.

How to use the Projectile Motion Calculator

Enter the launch speed $v_0$ in metres per second.
Enter the launch angle in degrees above the horizontal; use a negative angle for a downward throw and 0 for a horizontal launch.
Enter the launch height above the landing level if it is not zero, for example the height of a cliff.
Change gravity only if the problem specifies another value; 9.81 m/s² is used when the box is empty.
Press Solve projectile to see the components, peak, flight time, range and impact velocity.

Formulas

Velocity components

v_x = v_0\cos\theta,\quad v_y = v_0\sin\theta

Position

x = v_x t,\quad y = h_0 + v_y t - \tfrac12 g t^2

Maximum height

H = h_0 + \frac{v_y^2}{2g}

Flight time

T = \frac{v_y + \sqrt{v_y^2 + 2gh_0}}{g}

Level-ground range

R = \frac{v_0^2\sin 2\theta}{g}

Worked example: 20 m/s at 45° from the ground

The worked example launches a ball at 20 m/s and 45° from the ground. Both components are $20\cos 45° = 20\sin 45° \approx 14.142$ m/s. The ball rises for $14.142/9.81 \approx 1.442$ s to a maximum height of about $10.19$ m.

The total flight time is twice the rise time, about $2.883$ s, so the range is $14.142 \times 2.883 \approx 40.77$ m, matching $\frac{20^2 \sin 90°}{9.81}$ . The ball lands at 20 m/s, the same speed it was launched with, because it returns to the starting height.

The solution below is generated by the same engine that powers the calculator above.

Final answer

T \approx 2.8832081\ \text{s},\ R \approx 40.77472\ \text{m},\ H \approx 10.19368\ \text{m}

How we read your problem

Treat horizontal and vertical motion separately: constant velocity horizontally, constant downward acceleration g vertically, ignoring air resistance.

v_0 = 20\ \text{m/s},\ \theta = 45^\circ,\ h_0 = 0\ \text{m}

Step-by-step solution

Split the launch velocity into components
$v_x = v_0\cos\theta = 20\cos 45^\circ = 14.142136,\quad v_y = v_0\sin\theta = 14.142136\ \text{m/s}$
Time to the highest point
$t_{\text{peak}} = \frac{v_y}{g} = 1.441604\ \text{s}$
Maximum height
$H = h_0 + \frac{v_y^2}{2g} = 10.19368\ \text{m}$
Time of flight (land at height 0)
$h_0 + v_y t - \tfrac12g t^2 = 0 \Rightarrow t = \frac{v_y + \sqrt{v_y^2 + 2gh_0}}{g} = 2.8832081\ \text{s}$
Horizontal range
$R = v_x t = 14.142136 \times 2.8832081 = 40.77472\ \text{m}$
Impact velocity
$v_y(T) = v_y - gT = -14.142136,\quad |v| = \sqrt{v_x^2 + v_y(T)^2} = 20\ \text{m/s at } 45^\circ \text{ below horizontal}$

Formulas used

Components
$v_x = v_0\cos\theta,\ v_y = v_0\sin\theta$
Height
$y = h_0 + v_y t - \tfrac12 g t^2$
Range on level ground
$R = \frac{v_0^2 \sin 2\theta}{g}$

Without air resistance the horizontal velocity never changes, while the vertical velocity falls by g every second. The flight ends when the height returns to zero.

Verification

Energy check: v² at impact = 400 and v₀² + 2gh₀ = 400 (equal).
Level-ground formula: v₀² sin 2θ / g = 40.77472 m.

Detailed explanation

Launching from a height

When the launch point is above the ground, the projectile falls further than it rose, so the flight lasts longer than twice the rise time. The landing time comes from solving $h_0 + v_y t - \frac12 g t^2 = 0$ and keeping the positive root. A ball thrown at 20 m/s and 30° from a 10 m platform stays in the air about 2.774 s and travels about 48.04 m.

A horizontal launch, $\theta = 0$ , is the simplest case: the vertical motion is pure free fall. An object launched horizontally at 15 m/s from 20 m falls for $\sqrt{2 \times 20 / 9.81} \approx 2.019$ s and lands about 30.29 m away, regardless of how fast gravity is pulling it sideways, because it is not.

Impact speed and energy

At landing, the horizontal component is unchanged and the vertical component is $v_y - gT$ . Combining them with Pythagoras gives the impact speed, and the arctangent of their ratio gives the angle below the horizontal.

Energy conservation provides an independent check: without air resistance, $v_{\text{impact}}^2 = v_0^2 + 2gh_0$ . The calculator compares the two and reports the check, so you can see that the step-by-step kinematics and the energy method agree.

Limits of the model

Real projectiles experience air resistance, which shortens the range and makes the descending path steeper than the ascending one. For dense, slow objects over short distances the ideal model is accurate to a few percent, but for a shuttlecock or a long-range shot it can be far off.

The model also assumes level landing ground and a uniform gravitational field. These are the assumptions used in school and first-year university physics, and they are the ones this calculator follows.

Common mistakes

Using the level-ground range formula when the launch height is not zero.
Mixing up sine and cosine for the components; the vertical component uses sine of the angle above the horizontal.
Using the total speed instead of the vertical component when finding the maximum height.
Leaving a handheld calculator in radian mode with angles in degrees.
Assuming the horizontal velocity changes during flight when air resistance is ignored.

Tips

Draw the velocity triangle at launch before calculating anything.
At the highest point the vertical velocity is zero, but the horizontal velocity is not.
For maximum range on level ground, aim at 45°; from a height, the best angle is slightly less.
Keep full precision for the flight time because the range multiplies it.
Use the kinematics calculator for the vertical motion alone if a question asks about a particular height.

Frequently asked questions

What angle gives the maximum range?

On level ground and without air resistance, 45° gives the greatest range because sin 2θ reaches its maximum of 1 there. Launching from a height lowers the best angle slightly below 45°.

How do you calculate the time of flight of a projectile?

Solve h₀ + v_y t − ½gt² = 0 for the positive time. On level ground this simplifies to T = 2v₀ sin θ / g.

How do you find the maximum height of a projectile?

At the top the vertical velocity is zero, so v_y² = 2g(H − h₀), giving H = h₀ + (v₀ sin θ)² / (2g).

Why do 30° and 60° give the same range?

The range depends on sin 2θ, and sin 60° equals sin 120°. Any two angles that add to 90° give the same range on level ground, although the steeper launch flies higher and longer.

Does the mass of the projectile matter?

Not in the ideal model. Gravity gives every object the same acceleration, so mass cancels out. With air resistance, heavier compact objects are affected less.

Can I use this for a horizontal throw off a cliff?

Yes. Enter an angle of 0 and the cliff height as the launch height. The vertical motion is then pure free fall and the horizontal distance is speed times fall time.

Is air resistance included?

No. The calculator uses the standard ideal model with constant gravity and no drag, which is what most physics problems assume. Real ranges are usually somewhat shorter.

Launch your projectile

Enter the speed, angle and height above to get the flight time, range, peak height and impact velocity.

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