An online calculator to calculate the maximum height, range, time of flight, initial angle and the path of a projectile. The projectile equations and parameters used in this calculator are decribed below.
The vector initial velocity has two components:
V0x and V0y given by:
V0x = V0 cos(θ)
V0y = V0 sin(θ)
The vector acceleration A has two components Ax and Ay given by: (acceleration along the y axis only)
Ax = 0 and Ay = - g = - 9.8 m/s2
At time t, the velocity has two components given by
Vx = V0 cos(θ)
and
Vy = V0 sin(θ) - g t
The displacement is a vector with the components x and y given by:
x = V0 cos(θ) t
and
y = y0 + V0 sin(θ) t - (1/2) g t2
The time Tm at which y is maximum is at the vertex of y = y0 + V0 sin(θ) t - (1/2) g t2 and is given by
Tm = V0 sin(θ) / g
Hence the maximum height ymax reached by the projectile is given by
ymax = y(Tm) = y0 + V0 sin(θ) Tm - (1/2) g (Tm)2
The time of flight Tf is found by solving the equation
y0 + V0 sin(θ) t - (1/2) g t2 = 0
for t and taking the largest positive solution.
The shape of the trajectory followed by the projectile is found as follows
Solve the formula \( \; x = V_0 cos(\theta) t \; \) for \( t \) to obtain
\[ t = \dfrac{x}{V_0 cos(\theta)} \]
Substitute for t in y and simplify to obtain
\[ y = - \dfrac{g \; x^2}{2( V_0 \cos(\theta))^2} + x \tan(\theta) + y_0\]
The equation of the path of the projectile is a parabola of the form
\( y = A x^2 + B x + C\)
Horizontal Range = \( x(T_f) = V_0 cos(\theta) T_f \)