PTT

Physics

Projectile Trajectory Table Calculator

Calculate a complete constant-gravity, no-drag trajectory from speed, angle, launch height, gravity, and an entered number of equal time samples.

Launch speed (m/s)Launch angle (degrees)Launch height (m)Gravitational acceleration (m/s²)Displayed trajectory points

Horizontal speed component (m/s)-

Vertical speed component (m/s)-

Ideal time until landing (s)-

Ideal horizontal range (m)-

Time to apex (s)-

Ideal apex height (m)-

Time interval between table points (s)-

Decision view

Ideal projectile trajectory

Ideal projectile trajectoryHorizontal distance in meters is the x-axis; height above landing in meters is the y-axis for every calculated table point.

Exact scenario comparisonLaunch speed (m/s) changes while all other entered assumptions remain constant.

Launch speed (m/s)	Horizontal speed component (m/s)	Vertical speed component (m/s)	Ideal time until landing (s)	Ideal horizontal range (m)	Time to apex (s)	Ideal apex height (m)	Time interval between table points (s)

Period-by-period detail

Complete ideal projectile trajectory table

Every row advances by an equal time interval and recalculates horizontal distance, height, and vertical velocity.

How to use Projectile Trajectory Table Calculator

Enter launch speed, angle, height, and gravity.
Choose at least three trajectory points.
Read flight time, range, and apex.
Inspect the curve and full time table.

Calculator guide

Understanding Projectile Trajectory Table Calculator

A projectile path is a time-dependent curve, not a single range value. This calculator resolves the launch velocity and samples horizontal position, height, and vertical velocity from launch through the ideal landing time.

Components Speed splits into x and y.

Time root Vertical motion sets landing.

Function Every time maps to x and y.

Check Final height returns to zero.

Calculation method

How the calculation works

Resolve the entered projectile launch velocity, solve the ideal landing time from vertical motion, and sample the complete no-drag trajectory at equal time intervals. Resolve the entered speed into horizontal and vertical components, solve the positive root of the vertical position equation for landing time, then substitute each sample time into the position equations.

Detailed calculation process

Resolve and sample the complete ideal projectile path

The default launches at 32 m/s and 42 degrees from 1.5 m above the landing level under g = 9.80665 m/s², using 13 plotted points.

General formula: v_x = v_0 cos(theta)v_y = v_0 sin(theta)t_f = [v_y + sqrt(v_y^2 + 2gh_0)]/gx(t) = v_x ty(t) = h_0 + v_y t - gt^2/2t_apex = v_y/gy_apex = h_0 + v_y^2/(2g) Horizontal speed stays constant in this ideal model. Vertical speed decreases by g each second; the positive quadratic root gives landing time, and substituting time from zero to that root traces the curve.

What each symbol means

v_0 Entered launch speed (m/s).

theta Launch angle above horizontal (degrees; converted for trigonometry).

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

h_0 Launch height above the landing level (m).

g Gravitational acceleration (m/s²).

t_f Ideal time until y returns to zero (s).

x(t), y(t) Horizontal distance and height at time t (m).

t_apex, y_apex Time and height at the trajectory apex (s and m).

Worked substitution with the default inputs

1. Resolve launch velocity theta = 42 degrees = 0.733038 radv_x = 32 cos(42 degrees) = 23.780634 m/sv_y = 32 sin(42 degrees) = 21.412179 m/s The two perpendicular components reconstruct the entered 32 m/s speed.

2. Solve the landing time t_f = [21.412179 + sqrt(21.412179^2 + 2(9.80665)(1.5))]/9.80665t_f = 4.435834 s The negative quadratic root describes a time before launch and is discarded.

3. Calculate range R = v_x t_fR = 23.780634(4.435834) = 105.486941 m With no horizontal acceleration, distance is speed multiplied by flight time.

4. Locate the apex t_apex = 21.412179/9.80665 = 2.183435 sy_apex = 1.5 + 21.412179^2/[2(9.80665)] = 24.876047 m At the apex the vertical velocity is zero.

5. Set the table interval and reconcile Delta_t = 4.435834/(13-1) = 0.369653 sy(t_f) = 1.5 + 21.412179(4.435834) - 9.80665(4.435834)^2/2 = 0 m Thirteen points create twelve equal intervals and the last sampled height returns to the landing level.

The default ideal path remains airborne for 4.436 s, reaches 24.876 m, and lands 105.487 m horizontally from launch.

Purpose-built visual

Read the trajectory function curve

The live curve plots the actual parabola, labels launch and apex, and marks the calculated landing point.

Curve Ideal height by horizontal distance.

Apex Maximum modeled height.

Landing Positive-root endpoint.

Cards Components, time, and range.

Worked situations

Practical examples

At 42 degrees, horizontal speed is 23.781 m/s.
The default apex occurs at 2.183 s.
The last table row reconciles to zero height.

Better inputs

Useful tips

Use a consistent landing-height reference.
Increase points for a smoother display.
Treat the curve as an ideal classroom model.

Before relying on the result

Limitations and common mistakes

Air drag, lift, wind, spin, curvature, and obstacles are excluded.
The landing surface is assumed horizontal at y = 0.
Do not use the result for real-world targeting or safety decisions.

Reference

Key terms

Trajectory: Position of the projectile through time.
Apex: Highest point of the ideal path.
Velocity component: Horizontal or vertical part of launch velocity.
Positive root: Physically relevant landing time after launch.

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 motion is uniform while vertical displacement contains a squared-time gravity term.

Why are there two quadratic roots?

One lies before launch; the positive root is the future landing.

Does point count change the answer?

No. It only changes table and curve sampling.

Is air resistance included?

No. This is an ideal constant-gravity model.