Projectile Motion Calculator

What is Projectile Motion?

Throw a ball, kick a soccer ball, or launch a rocket. If you ignore air resistance, these objects all follow a specific curved path called a Parabola. This movement is known as Projectile Motion.

The fascinating thing about projectile motion is that it is a combination of two independent motions happening at the same time:

• Horizontal Motion (X-axis): The object moves at a constant speed. (Gravity doesn't pull sideways).
• Vertical Motion (Y-axis): The object constantly changes speed due to gravity pulling it down (9.81 m/s²).

Our Projectile Motion Calculator does the complex vector trigonometry for you, instantly telling you where the object will land and how high it will go.

Breaking Down Velocity (Vectors)

If you launch a projectile at an angle, the first step is to break the initial velocity (v₀) into its horizontal and vertical parts. We use trigonometry for this.

1. Calculating Max Height (H)

How high does the projectile go? The maximum height occurs when the vertical velocity hits zero (just before it starts falling back down). The formula depends only on the vertical launch speed and gravity.

H = h₀ + (v_y²) / (2g)

• h₀ = Initial Height (if launched from a cliff).
• v_y = Initial Vertical Velocity.
• g = Gravity (9.81 m/s²).

2. Calculating Time of Flight (t)

How long does it stay in the air? The time depends purely on gravity and how high you threw it. If you are launching and landing on the same level ground, the formula is simple:

t = (2 · v_y) / g

If you launch from a height (h₀ > 0), the math gets harder because the fall takes longer than the rise. Our calculator handles this using the quadratic formula automatically.

3. Calculating Range (R)

The Range is how far away the projectile lands. Since horizontal speed is constant, range is simply:

Distance = Speed × Time

R = v_x · t

The "45 Degree" Rule

If you want to throw a ball as far as possible, at what angle should you throw it?

Physics tells us that the optimal angle for maximum range (on level ground) is 45 degrees.
If you throw lower (< 45°), the ball moves fast horizontally but hits the ground too soon.
If you throw higher (> 45°), the ball stays in the air for a long time but spends too much energy going up rather than forward.

Real World Examples

1. Sports (Soccer / Golf)

A soccer goalkeeper kicking a goal kick wants maximum distance, so they aim for 45°. A golfer hitting a "flop shot" wants maximum height and minimum distance to stop the ball on the green, so they use a high angle (60° wedge).

2. Fireworks

Pyrotechnicians need to calculate exactly how high a shell will go so it explodes at a safe altitude. They calculate the fuse timing based on the "Time of Flight" formula to ensure the burst happens at the peak of the parabola.