A well-known conclusion from elementary ballistic trajectory analysis is that the horizontal range of a projectile is maximized for a launch angle of 45°. The derivation of this result assumes that the launch speed is the same for all possible launch angles, but this neglects the fact that the work done against gravity by the launching mechanism subtracts from the kinetic energy imparted to the projectile. Here we derive a formula for the optimum launch angle for a spring-powered device that takes this angle-dependent factor into account, and examine the resulting consequences for the range of the projectile.

1.
E.g.,
D. C.
Giancoli
,
Physics: Principles with Applications
, 7th ed. (
Pearson
,
New York
,
2014
), p.
63
.
2.
Bradley
Allen
, “
An introductory calculus-based mechanics investigation
,”
Phys. Teach.
55
,
500
501
(
Nov.
2017
).
3.
Jeffrey W.
Schnick
, “
Projectile motion details
,”
Phys. Teach.
32
,
266
269
(
May
1994
).
AAPT members receive access to The Physics Teacher and the American Journal of Physics as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.