Ratio of the maximum height to the range of a projectile

Help with physics problem and solution

Find the formula for the ratio of the maximum height H to the range of a projectile. Does it depend on the initial velocity and gravitational acceleration g?

From the chapter “Projectile Motion” we know that the maximum height of a projectile is

H = (1/2g)v_o²sin²α        (1)

and the range is given by

R = (1/g)2v_o²sinαcosα       (2)

Dividing equation (1)  by equation (2) we get

 

H/R = (1/4)(sinα/cosα) = (1/4)tanα                (4)

tanα is the tangent of angle α. Depending on the country it is written as

tanα  or  tgα

 

From formula (4) we see that this ratio does not depend on the initial velocity v_o nor on the gravitational acceleration g, because these two parameters are NOT present in the formula.

This is the general way of reasoning in physics or mathematics. If a parameter is not present in the equation for some quantity – this quantity does not depend on it.

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