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Horizontal Distance
The horizontal distance traveled by a projectile depends on its initial velocity, launch angle, and time. Since there is no horizontal acceleration (ignoring air resistance), horizontal motion is uniform.
x = v₀ cosθ × t
Vertical Distance
The vertical distance accounts for both the upward component of the initial velocity and the downward pull of gravity. The trajectory reaches its peak when the vertical velocity is zero.
y = v₀ sinθ × t − ½gt²
Horizontal Velocity
The horizontal component of velocity remains constant throughout the flight because gravity acts only vertically. It depends solely on the initial speed and launch angle.
vₓ = v₀ cosθ
Vertical Velocity
The vertical velocity decreases as the projectile rises (gravity decelerates it) and increases as it falls. It equals zero at the apex of the trajectory.
vᵧ = v₀ sinθ − gt
Range (Equal Elevation)
The range formula gives the total horizontal distance when launch and landing elevations are equal. Maximum range occurs at a 45° launch angle in a vacuum.
R = v₀² sin(2θ) / g
How It Works
Projectile motion splits into two independent parts: horizontal (constant velocity, x = v₀cosθ·t) and vertical (constant acceleration due to gravity, y = v₀sinθ·t − ½gt²). Together they trace a parabolic trajectory. This calculator solves for distance, velocity, angle, or time in either direction.
Example Problem
A ball is launched at 20 m/s at 45°. How far does it travel horizontally in 2 seconds?
- vₓ = 20 × cos(45°) = 14.14 m/s
- x = 14.14 × 2 = 28.28 m
When to Use Each Variable
- Solve for Horizontal Distance — when you know the launch velocity, angle, and flight time, e.g., calculating how far a ball travels before landing.
- Solve for Vertical Distance — when you need the height at a specific time, e.g., determining if a projectile clears a wall at a given range.
- Solve for Horizontal Velocity — when you need the constant horizontal speed component, e.g., analyzing the forward speed of a launched object.
- Solve for Vertical Velocity — when you need the instantaneous vertical speed at a given time, e.g., finding the velocity just before impact.
- Solve for Range — when launch and landing elevations are equal, e.g., calculating total distance for an artillery shell on flat terrain.
Key Concepts
Projectile motion decomposes into two independent components: constant horizontal velocity (no horizontal forces, ignoring air resistance) and uniformly accelerated vertical motion (gravity at 9.81 m/s² downward). The resulting trajectory is a parabola. Maximum range on level ground occurs at 45°, and the time of flight depends only on the vertical component.
Applications
- Sports science: analyzing basketball shot trajectories, football punt distances, and golf ball flight paths
- Military ballistics: computing artillery range tables and mortar firing angles for given distances
- Civil engineering: designing water fountain arcs and drainage outfall trajectories
- Space exploration: calculating launch angles and velocities for suborbital rocket trajectories
Common Mistakes
- Forgetting to decompose velocity into horizontal and vertical components — you must use v₀cosθ and v₀sinθ separately, not the full velocity in both equations
- Using the range formula for unequal launch and landing elevations — R = v₀²sin(2θ)/g only works when the projectile lands at the same height it was launched
- Ignoring air resistance for high-speed or long-range calculations — drag significantly reduces range and alters the optimal launch angle below 45°
Frequently Asked Questions
What launch angle gives the maximum range?
On flat ground with no air resistance, 45° maximizes range. With air resistance or from an elevated launch point, the optimal angle is typically less than 45°.
Does air resistance affect projectile motion?
Yes, significantly. Air resistance reduces range and maximum height, and the trajectory is no longer a perfect parabola. These equations assume a vacuum (no air resistance).
Why does horizontal velocity stay constant?
Gravity acts only vertically. With no horizontal force (ignoring air resistance), Newton’s first law says the horizontal component of velocity does not change.
Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
Related Calculators
- Constant Acceleration Calculator — solve the underlying 1-D kinematics equations.
- Gravity Equations Calculator — explore gravitational acceleration in detail.
- Force Equation Calculator — find the force behind the acceleration.
- Kinetic Energy Calculator — find the energy of the projectile at any point.
- Potential Energy Calculator — calculate PE at the projectile's maximum height.
- Speed Unit Converter — convert initial velocity between m/s, ft/s, and mph.
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