![]() 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. It can find the time of flight, but also the components of velocity, the range of the projectile, and the maximum height of flight. 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. Projectile motion equations FAQ Our projectile motion calculator is a tool that helps you analyze parabolic projectile motion. 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. I tried to drive a formula, ending up having the horizontal distance traveled = v^2sin(2θ)/g. Example of Application A ball is thrown from the top of a building, 45 m high above ground level, with an initial velocity of 15 m s-1 in a horizontal direction ( Figure: 4 ). For the question of comparing the horizontal distance traveled of different initial angles of projection. The range (R) can be calculated using the formula: Figure 3: Projectile Motion C. A formula used to model the vertical motion of an object that is dropped, thrown straight up, or thrown straight down. ![]()
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