12.9: Summary
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This following table summarizes the formulæ for a projectile initially at the origin, fired with initial velocity v0 at an angle θ from the horizontal.
Table 9-1. Summary of formulæ for projectile motion.
Quantity | Formula |
---|---|
x(t) | x=(v0cosθ)t |
y(t) | y=−12gt2+(v0sinθ)t |
y(x) | y(x)=(−g2v20cos2θ)x2+(tanθ)x |
Time in flight | tf=2gv0sinθ |
Range at angle θ | R=v20gsin2θ |
Max. range (at θ=45∘) | Rmax=v20g |
Angle needed to hit target at range R for fixed v0 | θ=12sin−1(gRv20) |
Speed needed to hit target at range R for fixed θ | v0=√gRsin2θ |
Max. altitude | h=v20sin2θ2g |
Speed needed to hit target at (xt,yt) for fixed θ | (xt,yt) for fixed θ & v0=√gxt2(tanθ−ytxt)cos2θ |
Angle needed to hit target at (xt,yt) for fixed v0 | xtsin2θ−2ytcos2θ=gx2tv20 |
Note that this last expression must be solved iteratively for θ.