PTS

Physics

Projectile Trajectory Solver Calculator

Solve an ideal two-dimensional projectile trajectory, including velocity components, flight time, range, maximum height, target-position height, and a point-by-point path table.

Launch speed (m/s)Launch angle (degrees)Launch height (m)Gravity (m/s squared)Target horizontal distance (m)Trajectory table pointsComparison height (m)

Horizontal velocity-

Vertical velocity-

Positive flight time-

Modeled horizontal range-

Maximum future height (m)-

Time to apex-

Time to entered target distance-

Projectile height at target distance-

Target height minus comparison height-

Trajectory time step-

Decision view

Ideal projectile result geometry

Ideal projectile result geometryRange, future apex, target height, and comparison height use final result or prefixed input keys.

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

Launch angle (degrees)	Horizontal velocity	Vertical velocity	Positive flight time	Modeled horizontal range	Maximum future height (m)	Time to apex	Time to entered target distance	Projectile height at target distance	Target height minus comparison height	Trajectory time step

Period-by-period detail

Complete ideal projectile trajectory table

Every row advances exact time and recalculates horizontal position, height, vertical velocity, and distance remaining.

How to use Projectile Trajectory Solver Calculator

Enter launch speed, angle, starting height, and gravity.
Choose the horizontal target distance and comparison height.
Set the number of trajectory-table points.
Read the live trajectory curve and verify the target coordinate against the table.

Calculator guide

Understanding Projectile Trajectory Solver Calculator

An ideal projectile path is determined by the horizontal and vertical components of launch velocity, the starting height, and gravity. This solver keeps the component resolution, positive ground-intersection time, apex, target-position height, and complete trajectory table on the same coordinate system.

One coordinate system All outputs use the same origin, height, and gravity.

Positive root Only the forward-time ground intersection is reported.

Target is evaluated Height at the entered x-coordinate is solved directly.

Path is live Inputs redraw the function curve and table.

Calculation method

How the calculation works

Resolve launch velocity into horizontal and vertical components, solve the positive ground-intersection time, and generate an exact time-position trajectory table. For a downward launch, future apex height remains at launch height while the positive ground root controls flight time. Resolve the launch speed with sine and cosine, solve the positive quadratic root for ground impact, then evaluate horizontal position and vertical height at each trajectory time.

Detailed calculation process

Resolve the launch vector and solve the complete ideal trajectory

The default projectile starts at 32 m/s and 42 degrees from a height of 1.5 m under 9.80665 m/s² gravity.

General formula: v_x = v_0 cos(theta)v_y = v_0 sin(theta)t_f = [v_y + sqrt(v_y^2 + 2gh_0)]/gR = v_x t_fH_max = h_0 + v_y^2/(2g)y(x) = h_0 + (v_y/v_x)x - gx^2/(2v_x^2) The horizontal component remains constant in the ideal model. Gravity changes only vertical motion, so the positive root gives the impact time, horizontal speed converts that time to range, and the coordinate equation gives height at any entered horizontal distance.

What each symbol means

v_0 Entered launch speed (m/s).

theta Launch angle above the horizontal (degrees).

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

h_0 Launch height above the reference ground level (m).

g Entered gravitational acceleration (m/s²).

t_f, R Positive flight time (s) and horizontal range (m).

H_max, y(x) Maximum height and height at horizontal coordinate x (m).

Worked substitution with the default inputs

1. Resolve the launch velocity v_x = 32 cos(42°) = 23.7806 m/sv_y = 32 sin(42°) = 21.4122 m/s The two perpendicular components preserve the entered 32 m/s vector.

2. Solve the positive impact time t_f = [21.4122 + sqrt(21.4122^2 + 2(9.80665)(1.5))]/9.80665t_f = 4.43583 s The negative quadratic root represents a time before launch and is discarded.

3. Calculate range and apex R = (23.7806)(4.43583) = 105.4869 mH_max = 1.5 + 21.4122^2/[2(9.80665)] = 24.8760 m Range uses full flight time, while the apex follows from the vertical velocity reaching zero.

4. Evaluate the entered target distance t_85 = 85/23.7806 = 3.57434 sy(85) = 1.5 + 21.4122(3.57434) - 0.5(9.80665)(3.57434^2) = 15.3900 m The target coordinate is evaluated with the same component values rather than a separate approximation.

5. Reconcile the end point x(t_f) = 23.7806(4.43583) = 105.4869 my(t_f) = 1.5 + 21.4122(4.43583) - 0.5(9.80665)(4.43583^2) = 0 m Substituting the solved impact time must return ground height and the reported range.

The default ideal path remains aloft for 4.436 s, reaches 24.876 m, lands 105.487 m away, and is 15.390 m high at x = 85 m.

Mathematical trajectory

Follow the function curve from launch to ground

The chart plots the full parabolic path, marks the apex and entered target coordinate, and keeps reference ground visible.

Launch vector Component arrows explain the entered speed and angle.

Apex marker The peak matches maximum-height arithmetic.

Target coordinate The entered distance is plotted at its calculated height.

Impact check The curve returns to zero at the solved range.

Worked situations

Practical examples

At 32 m/s and 42°, horizontal velocity is 23.781 m/s.
The projectile crosses x = 85 m before impact and is 15.390 m high there.
The final trajectory row returns approximately zero height at 105.487 m.

Better inputs

Useful tips

Use one consistent metre-second unit system.
Increase trajectory points when a denser value table is needed.
Treat the target marker as a coordinate reference, not a safety recommendation.

Before relying on the result

Limitations and common mistakes

The model excludes air resistance, wind, spin, lift, changing gravity, terrain, and object dimensions.
Angles near vertical make horizontal target calculations numerically sensitive.
This idealized physics page is not a firing solution or safety tool.

Reference

Key terms

Velocity component: The horizontal or vertical share of the launch velocity vector.
Apex: The highest point where vertical velocity is zero.
Flight time: Positive elapsed time until the modeled path reaches reference ground.
Trajectory: The calculated x-y curve traced through time.

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 is the path parabolic?

Horizontal velocity is constant while vertical displacement contains a gravity-times-time-squared term.

Why use the positive quadratic root?

It is the ground intersection after launch; the other root occurs before time zero.

Can the target height be negative?

Yes, if the entered horizontal distance lies beyond the modeled impact point.

Does the solver include drag?

No. It is an ideal no-drag model.