Projectile Motion Simulator
Simulate projectile trajectories with adjustable launch angle, velocity, gravity, and launch height.
What is projectile motion?
Projectile motion describes the path of an object launched into the air and subject only to gravity (ignoring air resistance). The object follows a parabolic trajectory — moving forward at constant horizontal velocity while simultaneously accelerating downward at 9.81 m/s squared. This combination of constant horizontal motion and accelerating vertical motion creates the characteristic curved path.
The key insight is that horizontal and vertical motions are independent. The horizontal velocity remains constant (no horizontal force without air resistance), while the vertical velocity changes due to gravity. You can analyze each direction separately using kinematics equations, then combine the results to get the complete trajectory.
Key projectile equations
- Range: R = (v0 squared * sin(2*theta)) / g — Maximum horizontal distance.
- Maximum height: H = (v0 squared * sin squared(theta)) / (2g).
- Time of flight: T = (2 * v0 * sin(theta)) / g.
- Optimal angle for maximum range: 45 degrees (without air resistance).
How to use this tool
Enter the initial velocity and launch angle. The calculator shows the range, maximum height, time of flight, and the complete trajectory path. Adjust the angle to see how it affects the trajectory — notice that complementary angles (like 30 and 60 degrees) give the same range but different maximum heights.
Real-world considerations
In reality, air resistance significantly affects projectile motion, especially at high speeds. A baseball, football, or golf ball experiences drag that reduces range and changes the optimal launch angle to less than 45 degrees. The Magnus effect (spin-induced lift or curve) further complicates the trajectory. This calculator shows the ideal case without air resistance.
Frequently asked questions
Why is 45 degrees the optimal angle?
The range formula contains sin(2*theta), which is maximized when 2*theta = 90 degrees, meaning theta = 45 degrees. At this angle, the horizontal and vertical velocity components are equal, providing the best balance between going far and staying in the air long enough. With air resistance, the optimal angle drops to 35-42 degrees depending on the object.
Does the mass of the object affect projectile motion?
Without air resistance, no — all objects follow the same trajectory regardless of mass. This was famously demonstrated by Galileo (and later by astronaut David Scott on the Moon, dropping a hammer and feather). With air resistance, heavier objects are less affected by drag relative to their weight, so they travel farther.