Free Horizontal Projectile Motion Calculator

Understanding Horizontal Projectile Motion

This horizontal projectile motion calculator is designed to handle the specific scenario where an object is launched with a purely horizontal initial velocity from an elevated point. By entering just two known values—such as launch speed and height—you instantly obtain the projectile’s time of flight, horizontal range, and the complete trajectory. The tool complements a projectile range calculator and a time of flight calculator by focusing on the zero‑angle case, making it ideal for homework problems and quick physics checks.

In this type of motion, the launch angle is zero because the velocity is parallel to the ground. Consequently, the initial vertical velocity is zero, and the only acceleration acting on the projectile is gravity ($g \approx 9.8\ \text{m/s}^2$ downward). The motion is therefore a combination of constant‑velocity horizontal motion and uniformly accelerated vertical motion.

Key Equations for Horizontal Projectile Motion

To keep the mathematics simple, place the origin at the ground directly below the launch point. If the launch height is $h$ and the initial speed is $V$, then:

• Horizontal displacement (at time $t$):

$$x(t) = V t$$

• Vertical displacement (measured upward from the ground):

$$y(t) = h - \frac{1}{2} g t^{2}$$

The projectile hits the ground when $y = 0$. Solving $h - \frac{1}{2} g t^{2} = 0$ gives the time of flight:

$$t_{\text{flight}} = \sqrt{\frac{2h}{g}}$$

The horizontal range (also called the projectile range) follows directly:

$$R = V \cdot t_{\text{flight}} = V \sqrt{\frac{2h}{g}}$$

Eliminating $t$ between the $x$ and $y$ equations yields the trajectory equation (a parabola):

$$y = h - \frac{g}{2V^{2}} x^{2}$$

This is a much simpler expression than the general projectile trajectory formula, because the launch angle is fixed at $0^{\circ}$. The physics trajectory calculator embedded in this tool plots this parabolic curve for you.

Key Properties

• Horizontal velocity remains constant throughout the flight: $v_x = V$.
• Vertical velocity increases linearly downward: $v_y = -g t$.
• Horizontal acceleration is zero.
• Vertical acceleration is constant: $-g$.
• Because air resistance is neglected, mechanical energy is conserved (kinetic plus gravitational potential).

Note that the maximum height reached is simply the initial launch height $h$, since there is no upward vertical component.

Worked Example: Throwing a Ball from the Eiffel Tower

Suppose you throw a ball horizontally at $7\ \text{m/s}$ from the upper platform of the Eiffel Tower, which is $276\ \text{m}$ above the ground.

1. Set the velocity to $7\ \text{m/s}$.
2. Set the initial height to $276\ \text{m}$.
3. The horizontal projectile motion calculator instantly returns:
   • Time of flight: $\sqrt{2 \times 276 / 9.8} \approx 7.5\ \text{s}$
   • Horizontal range: $7 \times 7.5 \approx 52.5\ \text{m}$

You can also use the tool in reverse: for a desired range of $100\ \text{m}$, find the required launch speed from the same height.

Using the Calculator Effectively

The calculator accepts any two of the three main variables—velocity, height, and range—and fills in the missing ones. This flexibility makes it a practical horizontal velocity calculator and a powerful aid for exploring projectile motion. Whether you are a student verifying textbook problems or a hobbyist estimating real‑world throws, the tool automatically handles unit conversions and displays the trajectory graph.