What is the difference between the Range of a Projectile and the Maximum displacement of it?

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The first equation given is used to solve for the time of flight, $t_d$, on level ground and not the range, $d$.

As the 2nd Wikipedia link states here, the range in terms of this time of flight is

$$d = v_0\, t_d\, \cos \theta$$

which, after solving for $t_d$, yields

$$d = v_0\, \left( \frac{2 v_0}{g}\, \sin \theta \right) \cos \theta = \frac{v_0^2}{g}\sin 2\theta$$

The 2nd equation is generalized to un-level ground where the initial height is $y_0$ and the final height is $0$.

As the 1st Wikipedia link states in the opening section, you can verify that, when $y_0 = 0$, the second equation given simplifies to the equation above.

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Updated on June 16, 2022

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