Synchronous curves By eliminating from the ideal
Chapter , Problem 29PE(choose chapter or problem)
Synchronous curves By eliminating from the ideal projectile equations
\(x=\left(v_{0} \cos \alpha\right) t, y=\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2}\)
Show that \(x^{2}+\left(y+g t^{2} / 2\right)^{2}=v 0^{2} t^{2}\). This shows that projectiles launched simultaneously from the origin at the same initial speed will, at any given instant, all lie on the circle of radius centered at \(\left(0,-g t^{2} / 2\right)\), regardless of their launch angle. These circles are the synchronous curves of the launching.
Equation Transcription:
Text Transcription:
X = (v_0 cos alpha)t, y = (v_0 sin alpha)t - ½ gt^2
x^2 + (y+gt^2 /2)^2 = v0^2 t^2
(0, -gt^2 /2)
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