Projectile Motion

Splitting Launch Velocity Into Horizontal and Vertical Motion

Projectile motion describes the curved path of an object launched at an angle and moving under gravity alone, ignoring air resistance. Enter the launch speed, angle, and initial height, and this calculator finds the range, time of flight, and max height.

The Formula

$$\vA{v_x} = v_0 \cos(\theta), \quad \vB{v_y} = v_0 \sin(\theta)$$ $$\vC{t} = \frac{\vB{v_y} + \sqrt{\vB{v_y}^2 + 2gh_0}}{g}$$ $$\vD{\text{Range}} = \vA{v_x} \times \vC{t}$$

where $v_0$ is the launch speed, $\theta$ is the launch angle, $g$ is gravitational acceleration, and $h_0$ is the initial height.

Worked Example

An object launched at 20 m/s at a 30° angle from ground level:

1. Velocity components: $\vA{v_x} = 20\cos(30°) \approx \vA{17.32}$ m/s, $\vB{v_y} = 20\sin(30°) = \vB{10}$ m/s.
2. Time of flight: $\vC{t} = \frac{10 + \sqrt{10^2}}{9.80665} \approx \vC{2.04}$ s, so range $\vD{17.32 \times 2.04} \approx \vD{35.3}$ m.

Key Factors to Consider

• A launch angle of 45 degrees maximizes range on level ground, when launching from and landing at the same height. This is a well-known result of projectile motion — any angle above or below 45 degrees (from a level launch) produces a shorter range for the same launch speed, which is why 45 degrees is the classic textbook answer for “farthest throw” questions.
• Horizontal and vertical motion are independent of each other throughout the flight. Gravity only ever acts on the vertical component of velocity — the horizontal velocity stays constant the entire time (ignoring air resistance), which is exactly why the horizontal and vertical velocity components can be calculated separately and then combined.
• This model ignores air resistance entirely, which is a real simplification for many practical scenarios. Air resistance meaningfully affects lighter or less aerodynamic objects (like a ball with a lot of surface area relative to its mass) more than it affects dense, compact ones — this calculator’s results are most accurate for the latter and increasingly approximate as air resistance becomes more significant.
• A nonzero launch height changes the optimal angle for maximum range away from the classic 45 degrees. Launching from an elevated point (like a cliff) actually favors a launch angle slightly LESS than 45 degrees for maximum range, since the projectile has extra time in the air to travel horizontally before it needs to fall back down to ground level.

Common Mistakes

• Using the full launch speed directly in the range formula instead of the horizontal component. Range depends on horizontal velocity ($v_0 \cos(\theta)$), not the raw launch speed — skipping the decomposition step overstates the range at any angle other than 0°.
• Assuming 45 degrees is always the optimal launch angle. That’s only true when launch and landing height are equal — launching from an elevated starting height shifts the optimal angle below 45 degrees, as noted above.
• Treating this calculator’s result as an exact real-world prediction. The model ignores air resistance entirely, so a light or less aerodynamic object (a beach ball, a badminton shuttlecock) will fall noticeably short of the calculated range and time of flight in practice.

Useful to Know

• Modeling a straight vertical drop instead of an angled launch? Free Fall Calculator calculates fall time and impact velocity under gravity alone.
• Need the force behind the launch itself, not just the resulting trajectory? Force Calculator (F = ma) calculates force from mass and acceleration.
• Curious how the object’s energy shifts between height and speed during the flight? Kinetic & Potential Energy Calculator calculates kinetic and potential energy.