As a simple example, suppose we observe a mass m in circular orbit of radius R around a parent mass M, so that the orbit equation is r(θ)=R (a constant), and so u=1/R. The p...As a simple example, suppose we observe a mass m in circular orbit of radius R around a parent mass M, so that the orbit equation is r(θ)=R (a constant), and so u=1/R. The polar equation of a circle passing through the origin is r=2acosθ, where a is the radius of the circle. \frac{d^{2} u}{d \theta^{2}} & =\frac{2 a \cos ^{3} \theta+4 a \cos \theta \sin ^{2} \theta}{4 a^{2} \cos ^{4} \theta} \\ \notag\\
The difference between the two paths is due to air resistance acting on the object, →Fair=−bv2ˆv, where ˆv is a unit vector in th...The difference between the two paths is due to air resistance acting on the object, →Fair=−bv2ˆv, where ˆv is a unit vector in the direction of the velocity. (For the orbits shown in Figure 5.1,b=0.01N⋅s2⋅m−2, |→v0|=30.0m⋅s, the initial launch angle with respect to the horizontal θ0=21∘ an…