![]() ![]() ![]() A more realistic scenario is having the direction of gravity towards a center, which is definitely much harder to derive such an equation, and also you will have to redefine the distance traveled as Δθr, assuming that Earth is a perfect sphere with radius(r). However, this only works for the scenario that the direction of gravity is always one direction that is vertically downwards. Hence the equation can be simplified to s = v^2sin(2θ)/g. Lets remind us about the trigonometry identity sin(2θ) = 2cos(θ)sin(θ). Subsititing the equation, getting s = 2v^2sin(θ)cos(θ)/g. From the equation s = vcos(θ)t, and t = 2vsin(θ)/g. Rearranging the equation for finding t, vsin(θ)/g = t, this is the time it takes to reach its maximum height, so we multiply by 2 to get the total time for it to reach the maximum height and return back to the initial height. At maximum height, the vertical velocity(vsin(θ)) is reduced to zero, so the equation should give vsin(θ) - gt = 0. Knowing that the time it takes for the projectile to reach the maximum height from its initial height is the same as the time it takes to fall from the maximum height back to its initial height. So the issue is to find time(t), the time is affected by the vertical component of velocity and the acceleration due to gravity(g). Knowing that the horizontal velocity = vcos(θ), so we can get the horizontal distance(s) = horizontal velocity x time, s = vcos(θ)t.Ģ. ![]() Hence the optimal angle of projection for the greatest horizontal distance is 45° because sin(90) = 1, and any other angle will result in a value smaller than 1.ġ. It can be found by using an equation with a vertical velocity formula from among the list of classic Newtonian projectile motion physics equations, or an online calculator. ![]() I tried to drive a formula, ending up having the horizontal distance traveled = v^2sin(2θ)/g. Vertical velocity is that component of an objects displacement in space over a given time t in the y-direction only. The time of flight T f is found by solving the equationįor t and taking the largest positive solution.For the question of comparing the horizontal distance traveled of different initial angles of projection. Hence the maximum height y max reached by the projectile is given by The time T m at which y is maximum is at the vertex of y = y 0 + V 0 sin(θ) t - (1/2) g t 2 and is given by The displacement is a vector with the components x and y given by: V x = V 0 cos(θ) and V y = V 0 sin(θ) - g t The vector acceleration A has two components A x and A y given by: (acceleration along the y axis only)Īt time t, the velocity has two components given by The vector initial velocity has two components: V 0x and V 0y given by: Projectile Equations used in the Calculator and Solver Range = 50m, Initial Velocity: V 0 = 30m/s, Initial Height: y 0 = 10mĭecimal Places = 4 Initial Angle = ° Maximum Height = meters Flight Time= seconds Equation of the Path:: y = x 2 + x + The outputs are the initial angle needed to produce the range desired, the maximum height, the time of flight, the range and the equation of the path of the form \( y = A x^2 + B x + C\) given V 0 and y 0. Initial Velocity: V 0 = 30m/s, Initial Angle: θ = 50°, Initial Height: y 0 = 10mĭecimal Places = 4 Maximum Height = meters Flight Time= seconds Range = meters Equation of the Path: y = x 2 + x +Ģ - Projectile Motion Calculator and Solver Given Range, Initial Velocity, and Height Enter the range in meters, the initial velocity V 0 in meters per second and the initial height y 0 in meters as positive real numbers and press "Calculate". The outputs are the maximum height, the time of flight, the range and the equation of the path of the form \( y = A x^2 + B x + C\). The projectile equations and parameters used in this calculator are decribed below.ġ - Projectile Motion Calculator and Solver Given Initial Velocity, Angle and Height Enter the initial velocity V 0 in meters per second (m/s), the initial andgle θ in degrees and the initial height y 0 in meters (m) as positive real numbers and press "Calculate". An online calculator to calculate the maximum height, range, time of flight, initial angle and the path of a projectile. ![]()
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