What is Projectile Motion?
Projectile motion is the path an object takes when it is thrown, shot, or launched near the Earth's surface. The object, or "projectile," moves in a curved path (a parabola) under the influence of gravity alone. This calculation assumes that air resistance is negligible.
The key principle is to analyze the horizontal and vertical components of motion separately.
- Horizontal Motion: There is no acceleration, so the horizontal velocity ($v_x$) is constant.
- Vertical Motion: The object is only affected by gravity ($g$), which causes a constant downward acceleration.
Key Formulas
To solve for the projectile's path, we first break the initial velocity ($v_0$) and launch angle ($\theta$) into horizontal ($v_{0x}$) and vertical ($v_{0y}$) components:
1. Time of Flight ($T$)
The total time the object is in the air. We find this by solving the vertical position equation $y(t) = y_0 + v_{0y}t - \frac{1}{2}gt^2$ for $t$ when the object hits the ground ($y(t) = 0$). This requires the quadratic formula, which gives:
2. Maximum Height ($H$)
The peak of the projectile's path. This occurs when the vertical velocity becomes zero. The formula, which includes the initial height $y_0$, is:
3. Horizontal Range ($R$)
The total horizontal distance traveled. Since horizontal velocity $v_{0x}$ is constant, we simply multiply it by the total time of flight $T$:
Note: If the launch height $y_0 = 0$, the formulas simplify to the more common versions: $T = \frac{2v_0 \sin(\theta)}{g}$, $H = \frac{(v_0 \sin(\theta))^2}{2g}$, and $R = \frac{v_0^2 \sin(2\theta)}{g}$. Our calculator handles both cases.