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.

1 - Projectile Motion Calculator and Solver Given Initial Velocity, Angle and Height

2 - Projectile Motion Calculator and Solver Given Range, Initial Velocity, and Height

Projectile Equations used in the Calculator and Solver

The vector initial velocity has two components: V_0x and V_0y given by:
V_0x = V₀ cos(θ)
V_0y = V₀ sin(θ)
The vector acceleration A has two components A_x and A_y given by: (acceleration along the y axis only)
A_x = 0 and A_y = - g = - 9.8 m/s²
At time t, the velocity has two components given by
V_x = V₀ cos(θ) and V_y = 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 T_m at which y is maximum is at the vertex of y = y₀ + V₀ sin(θ) t - (1/2) g t² and is given by
T_m = V₀ sin(θ) / g
Hence the maximum height y_max reached by the projectile is given by
y_max = y(T_m) = y₀ + V₀ sin(θ) T_m - (1/2) g (T_m)²
The time of flight T_f 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 = \( x(T_f) = V_0 cos(\theta) T_f \)

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