Projectile Trajectory Calculator

Projectile Motion Basics

A projectile launched with initial speed $v$ at angle $\theta$ above horizontal follows a parabolic path under uniform gravity. Air resistance is neglected. Launch and landing heights can differ when the projectile starts above ground level.

Key Equations

$$R = \frac{v^2 \sin(2\theta)}{g}$$ $$H_{\max} = h_0 + \frac{(v\sin\theta)^2}{2g}$$ $$t = \frac{v\sin\theta + \sqrt{(v\sin\theta)^2 + 2gh_0}}{g}$$

Range $R$ is horizontal distance when landing at the same elevation as the launch point on level ground ($h_0 = 0$). Maximum height includes initial height $h_0$. Flight time $t$ covers ascent and descent to ground level.

Optimal Launch Angle

On level ground, maximum range occurs at $\theta = 45°$ because $\sin(2\theta)$ peaks at 90°. From an elevated platform, the optimal angle is slightly below 45°.

Frequently Asked Questions

What value of g does this calculator use?

Standard gravity $g = 9.80665$ m/s² is used. Results are idealized and ignore air drag.

Why is 45 degrees often the best angle on flat ground?

Range is proportional to $\sin(2\theta)$. At 45°, $\sin(90°) = 1$, which is the maximum possible value.

How does initial height affect range?

The horizontal range formula assumes landing at the same height as launch. Initial height increases flight time and maximum altitude but does not change $R$ on level ground.

Can launch angle exceed 90 degrees?

No. Launch angle is limited to 0 through 90 degrees. Angles above 90° would point the velocity downward at launch.

Does air resistance change the trajectory?

Yes. Real projectiles slow down due to drag, reducing range and peak height. This tool models vacuum conditions for classroom physics problems.

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Trajectory Projectile Motion

Projectile Motion Basics

Key Equations

Optimal Launch Angle

Frequently Asked Questions

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