Relation between angles in projectile motion
Help with physics problem and solution
Find the relation between the elevation angle α of a projectile and the elevation angle β of the highest point of this projectile as seen from the launch point.
In such a type of a problem we can use all possible parameters that describe the subject, but in the final formula only the parameters given in the problem may be present. The figure below will help solve the problem.
From the figure we find
tanβ = H/(R/2) = 2H/R (1)
In the “Projectile Motion” { M2-1-projectile-motion.html} paragraph we can find the formulas for maximum height H and range R
(2)
(3)
The ratio
H/(R/2) = 2H/R
calculated from formulas (2) and (3) is, after a little bit of algebra
2H/R = (1/2)sinα/cosα (4)
As
sinα/cosα = tanα
we can write equation (4) in the form
2H/R = (1/2)tanα (5)
Comparing formula (5) and (1) we see that
tanβ = (1/2)tanα (6)
and this is the solution to this problem.
We made calculations using the initial speed of a projectile, but it is not present in the final formula (6) as it was not given in the problem. This formula tells us that the relation between the angles defined in the problem DOES NOT depend on the initial speed nor on gravitational acceleration g.