Trajectory of a Projectile
Trajectory of a projectile is the curved path given an initial velocity and is acted on by gravity. The projectile is acted upon by both vertical and horizontal motion along the trajectory.
Trajectory of a Projectile formula
| \(\large{ y = d \; tan \; \theta - \frac{ g \; d^2 }{ 2 \; v_0^2 \; cos^2 \; \theta } }\) |
Where:
\(\large{ y }\) = vertical position
\(\large{ \theta }\) = angle of the initial vertical from x-axis
\(\large{ g }\) = gravitational acceleration
\(\large{ d }\) = horizontal position
\(\large{ v_0 }\) = launch velocity
Horizontal range of a Projectile formula
| \(\large{ R = \frac{ v_0^2 \; sin \; 2 \theta }{ g } }\) |
Where:
\(\large{ R }\) = horizontal range
\(\large{ g }\) = gravitational acceleration
\(\large{ v_0 }\) = launch velocity
\(\large{ \theta }\) = vertical angle
Launch Velocity of a Projectile formula
| \(\large{ v_0 = \sqrt{ \frac{ R \; g }{ sin\; 2\theta } } }\) |
Where:
\(\large{ v_0 }\) = launch velocity
\(\large{ R }\) = horizontal range
\(\large{ g }\) = gravitational acceleration
\(\large{ \theta }\) = vertical angle
Maximum Height of a Projectile formula
| \(\large{ h_{max} = \frac{ v_0 \; sin^2 \; \theta }{ 2 \; g } }\) |
Where:
\(\large{ h_{max} }\) = maximum height
\(\large{ g }\) = gravitational acceleration
\(\large{ v_0 }\) = launch velocity
\(\large{ \theta }\) = vertical angle
Projectile Clearing an Object formula
| \(\large{ y_h = \frac{d \; v_{0y} }{v_{0x} } - \frac{1}{2} \; g \; \frac{x^2}{v_{0x}^2} }\) |
Where:
\(\large{ y_h }\) = vertical position
\(\large{ \theta }\) = angle of the initial vertical from x-axis
\(\large{ g }\) = gravitational acceleration
\(\large{ d }\) = horizontal position
\(\large{ v_0 }\) = launch velocity
Projectile Launch Angle formula
| \(\large{ \theta = \frac{ 1 }{ 2 } \; sin^{-1} \left( \frac{ g\; d }{ v_0^2 } \right) }\) |
Where:
\(\large{ \theta }\) = angle of the initial vertical from x-axis
\(\large{ g }\) = gravitational acceleration
\(\large{ d }\) = horizontal position
\(\large{ v_0 }\) = launch velocity
Time of flight of a Projectile formula
| \(\large{ t = \frac{ 2 \; v_0 \; sin \; \theta }{ g } }\) |
Where:
\(\large{ t }\) = time
\(\large{ g }\) = gravitational acceleration
\(\large{ v_0 }\) = launch velocity
\(\large{ \theta }\) = vertical angle
Trajectory of a Projectile on a Hill formulas
| \(\large{ d = v_{0d} \; t }\) | |
| \(\large{ h = v_{0h} \; t - \frac{1}{2} \; g \; t^2 }\) | |
| \(\large{ t = \frac{ 2 \; v_{0h} }{ g } \pm \sqrt{ \frac{ v_{0h}^2 }{ g^2 } - \frac{ 2 \;h }{ g } } }\) |
Where:
\(\large{ t }\) = time
\(\large{ g }\) = gravitational acceleration
\(\large{ v_0 }\) = launch velocity
\(\large{ \theta }\) = vertical angle