PLS

Physics

Projectile Launch Speed Calculator

Solve the ideal same-height range equation for launch speed, apply an entered margin, decompose velocity, and calculate reference flight time and apex height.

Target horizontal range (m)Launch angle (degrees)Launch height above landing (m)Gravitational acceleration (m/s²)Entered speed margin (%)

Required speed for same-height reference (m/s)-

Reference speed including entered margin (m/s)-

Horizontal speed component (m/s)-

Vertical speed component (m/s)-

Same-height flight-time reference (s)-

Height gained above launch (m)-

Apex above landing (m)-

Decision view

Launch-speed solution chain

Launch-speed solution chainTarget range and angle become required speed, components, flight time, and apex.

Exact scenario comparisonLaunch angle (degrees) changes while all other entered assumptions remain constant.

Launch angle (degrees)	Required speed for same-height reference (m/s)	Reference speed including entered margin (m/s)	Horizontal speed component (m/s)	Vertical speed component (m/s)	Same-height flight-time reference (s)	Height gained above launch (m)	Apex above landing (m)

How to use Projectile Launch Speed Calculator

Enter target range and launch angle.
Set gravity and launch-height reference.
Enter an optional speed margin.
Read the function curve, operating point, and velocity components.

Calculator guide

Understanding Projectile Launch Speed Calculator

For equal launch and landing height without drag, target range and launch angle determine a unique ideal speed. Resolving that speed into components exposes the flight-time and apex implications.

Inversion Range determines speed.

Angle sensitivity Required speed varies with angle.

Components Speed splits into x and y.

Reference path Time and apex follow.

Calculation method

How the calculation works

Solve the ideal no-drag projectile range equation for launch speed, then decompose the margin-adjusted speed into horizontal and vertical components. Invert the no-drag range equation, multiply by the entered speed margin, use cosine and sine for velocity components, then apply constant-gravity vertical-motion formulas.

Detailed calculation process

Solve launch speed from target range and angle

The default targets 80 m at 40 degrees under g = 9.80665 m/s2, with a 5% entered speed margin and zero launch-height offset.

General formula: v_0 = sqrt[Rg/sin(2theta)]v_r = v_0(1+m/100)v_x = v_r cos(theta)v_y = v_r sin(theta)t = 2v_y/gh_a = h_0+v_y^2/(2g) The same-height range equation is solved for initial speed. The margin-adjusted speed is then decomposed into horizontal and vertical components for the time and apex references.

What each symbol means

R Target same-height horizontal range (m).

g Gravitational acceleration (m/s2).

theta Launch angle above horizontal (degrees in input).

v_0 Required ideal same-height launch speed (m/s).

m, v_r Entered speed margin percent and recommended reference speed.

v_x, v_y Horizontal and vertical velocity components (m/s).

t Same-height reference flight time (s).

h_0, h_a Launch height offset and apex above landing (m).

Worked substitution with the default inputs

1. Convert and double the angle theta = 40 degrees = 0.698132 rad2theta = 80 degreessin(2theta) = 0.984808 Trigonometric evaluation uses the entered angle consistently.

2. Solve required speed v_0 = sqrt[80x9.80665/sin(80 degrees)]v_0 = 28.224717 m/s This solution assumes equal launch and landing height.

3. Apply the entered margin v_r = 28.224717(1+5/100) = 29.635953 m/s The margin is an explicit planning input, not a physical correction for drag.

4. Resolve velocity v_x = 29.635953 cos(40 degrees) = 22.702457 m/sv_y = 29.635953 sin(40 degrees) = 19.049624 m/s The orthogonal components reconstruct the margin-adjusted speed.

5. Calculate time and apex t = 2(19.049624)/9.80665 = 3.885042 sh_a = 0+(19.049624)^2/[2(9.80665)] = 18.502147 m The final figures are reference values under constant gravity and no drag.

The default same-height solution is 28.225 m/s; after the entered margin it is 29.636 m/s, with a 3.885 s reference flight time and 18.502 m apex.

Purpose-built visual

Read an angle-speed function curve and velocity triangle

A required-speed curve across launch angles highlights the entered operating point, while a vector triangle shows horizontal and vertical components.

Curve Required speed by angle.

Point Entered angle and speed.

Triangle Velocity components.

Apex card Vertical-motion result.

Worked situations

Practical examples

The ideal required speed is 28.225 m/s.
The 5% margin raises it to 29.636 m/s.
The adjusted vertical component gives an 18.502 m apex gain.

Better inputs

Useful tips

Keep the angle between zero and 90 degrees.
Use the equal-height assumption only where applicable.
Do not use this educational model for safety-critical targeting.

Before relying on the result

Limitations and common mistakes

Air resistance, wind, spin, lift, and shape are excluded.
The speed inversion is exact only for equal launch and landing height.
Real launch safety and terrain constraints are not modeled.

Reference

Key terms

Launch component: Horizontal or vertical part of velocity.
Range equation: Ideal same-height horizontal distance relationship.
Apex: Highest modeled trajectory point.
Speed margin: Entered multiplier above ideal reference.

Important note

Calculated from the entered values using the displayed physical model. Confirm that its assumptions, units, boundary conditions, and safety limits match the application.

Frequently asked questions

Why does required speed rise near 0 or 90 degrees?

The same-height range factor sin(2theta) approaches zero.

Does launch height affect the solved speed?

Not in this same-height inversion; it is used only in the apex-above-landing reference.

Is the margin a drag model?

No. It is only a user-entered multiplier.

Can this be used for real weapons or targeting?

No. It is an idealized educational mechanics calculation.