This Projectile Motion Calculator provides quick and accurate analysis of object trajectory, determining the Time of Flight, Maximum Height, and Horizontal Range based on the initial velocity and launch angle.
Projectile Motion Calculator
Projectile Motion Formulas
Formulas adapted from fundamental classical mechanics principles. Source 1: Wikipedia on Projectile Motion | Source 2: Khan Academy Physics
Variables Explained
- Initial Velocity ($v_0$): The speed at which the object is launched, measured in meters per second (m/s).
- Launch Angle ($\theta$): The angle above the horizontal axis at which the projectile is launched, measured in degrees ($0^\circ$ to $90^\circ$).
- Gravitational Acceleration ($g$): The downward acceleration due to gravity, typically $9.81 \text{ m/s}^2$ on Earth.
- Time of Flight ($T$): The total time the projectile remains in the air before hitting the ground (s).
- Maximum Height ($H$): The highest vertical position reached by the projectile during its flight (m).
- Horizontal Range ($R$): The total horizontal distance traveled by the projectile (m).
What is Projectile Motion?
Projectile motion is a form of motion experienced by an object or particle (a projectile) that is thrown near the Earth’s surface and moves along a curved path under the influence of gravity only. The primary assumption in this classical physics model is that air resistance is negligible, and the acceleration due to gravity ($g$) is constant both in magnitude and direction.
This calculator simplifies complex trajectory analysis into three main outputs: the Time of Flight (how long it travels), the Maximum Height (its peak altitude), and the Horizontal Range (how far it lands). Understanding these components is critical in fields like engineering, sports science (e.g., golf, baseball), and military ballistics.
How to Calculate Projectile Motion (Example)
Let’s use a sample scenario: A ball is launched with an Initial Velocity ($v_0$) of $30 \text{ m/s}$ at a Launch Angle ($\theta$) of $35^\circ$, assuming standard gravity ($g$) of $9.81 \text{ m/s}^2$. Here is the step-by-step calculation:
- Input Variables: $v_0 = 30$, $\theta = 35^\circ$, $g = 9.81$.
- Convert Angle: Convert $35^\circ$ to radians: $35 \times (\pi/180) \approx 0.6109$ radians.
- Calculate Time of Flight ($T$): $$T = \frac{2 \times 30 \times \sin(35^\circ)}{9.81} \approx 3.504 \text{ s}$$
- Calculate Maximum Height ($H$): $$H = \frac{(30 \times \sin(35^\circ))^2}{2 \times 9.81} \approx 16.71 \text{ m}$$
- Calculate Horizontal Range ($R$): $$R = \frac{30^2 \times \sin(2 \times 35^\circ)}{9.81} = \frac{900 \times \sin(70^\circ)}{9.81} \approx 86.13 \text{ m}$$
Frequently Asked Questions (FAQ)
Why is $45^\circ$ the optimal angle for maximum range? The formula for range is $R = (v_0^2 / g) \times \sin(2\theta)$. Since $v_0$ and $g$ are fixed, $R$ is maximized when $\sin(2\theta)$ is maximized. The maximum value of the sine function is 1, which occurs when its argument is $90^\circ$. Therefore, $2\theta = 90^\circ$, meaning $\theta = 45^\circ$.
Does this calculator account for air resistance? No. This calculator uses the simplified classical physics model, which assumes no air resistance (a perfect vacuum). In reality, air resistance (or drag) would reduce both the maximum height and the horizontal range.
What value of $g$ should I use if I’m not on Earth? The gravitational acceleration ($g$) changes depending on the celestial body. You would use the body’s surface gravity value. For example, on the Moon, $g \approx 1.625 \text{ m/s}^2$.
What is the difference between velocity and speed in this context? Speed is the magnitude of velocity. Initial Velocity ($v_0$) here refers to the initial speed, as the direction (the angle $\theta$) is specified separately. Velocity is a vector (magnitude and direction), while speed is a scalar (magnitude only).