Throw off a Cliff

Two-dimensional display of the trajectory of an object thrown from a cliff into the sea. Parameters of the trajectory that you can vary are the cliff height H, the initial velocity (Subscript[v, 0]), and the initial angle α.

THINGS TO TRY

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DETAILS

The trajectory of the object is given by

y(x) = H + x cot(α)- ((g )/(2 ((Subscript[v, 0])^2)(sin^2)α))(x^2).

For simplicity, set g=10 m/(s^2).

The main goal of the Demonstration is to explore the range of the trajectory, that is, the maximum x coordinate that can be achieved with a given height H and a given velocity (Subscript[v, 0]). It is well-known that in the case of a ground-to-ground throw (H=0) the range is maximized for α=45°, as shown by the snapshots. For H>0 this condition no longer holds and the general expression for the range is given by

(cos^2) (Subscript[α, opt]) = ((Subscript[v, 0]^2)/(2((Subscript[v, 0]^2) + g H))),

for which the maximal range becomes

range=(Subscript[x, max])=((Subscript[v, 0])/g)(Sqrt[(Subscript[v, 0]^2) + 2g H ]).

RELATED LINKS

Contributed by: Antoine Weis

Additional contributions by: Fernand Brunschwig

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