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A projectile is launched from the ground with initial velocity $\vec{u} = u_0\hat{i} + v_0\hat{j}$. Gravity acts in the $-y$ direction. The maximum displacement in the $x$-direction is:
Time of flight $= \dfrac{2v_0}{g}$ (time for vertical displacement to return to zero).
Horizontal range $= u_0 \times \dfrac{2v_0}{g} = \mathbf{\dfrac{2u_0 v_0}{g}}$
A projectile is thrown at angles $(45ยฐ-\theta)$ and $(45ยฐ+\theta)$ with the horizontal. The ratio of their horizontal ranges is:
Using $R = \dfrac{u^2\sin2\alpha}{g}$:
$R_1 = \dfrac{u^2\sin(90ยฐ-2\theta)}{g} = \dfrac{u^2\cos2\theta}{g}$
$R_2 = \dfrac{u^2\sin(90ยฐ+2\theta)}{g} = \dfrac{u^2\cos2\theta}{g}$
$R_1 : R_2 = \mathbf{1:1}$
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