Question

a connon ball has a range R on a horizontal plane.If h and h' are the greatest heights in the two paths for which this is possible ,then

Answer 1

Answer on Question #49632-Physics-Mechanics-Kinematics-Dynamics

A cannon ball has a range R on a horizontal plane. If h and h' are the greatest heights in the two paths for which this is possible, then

$$R = 4 \sqrt {h h ^ {\prime}}.$$

Solution

The two angles are complimentary.

$$h = \frac {u ^ {2} \sin^ {2} \theta}{2 g}; h ^ {\prime} = \frac {u ^ {2} \sin^ {2} (9 0 - \theta)}{2 g} = \frac {u ^ {2} \cos^ {2} \theta}{2 g}.$$

And

$$R = \frac {u ^ {2} \sin 2 \theta}{g} = \frac {2 u ^ {2} \sin \theta \cos \theta}{g}.$$

But

$$\sin \theta = \sqrt {\frac {2 g h}{u ^ {2}}}, \cos \theta = \sqrt {\frac {2 g h ^ {\prime}}{u ^ {2}}}.$$

Therefore

$$R = \frac {2 u ^ {2}}{g} \sqrt {\frac {2 g h}{u ^ {2}}} \sqrt {\frac {2 g h ^ {\prime}}{u ^ {2}}} = 4 \sqrt {h h ^ {\prime}}.$$

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