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 \(r(\theta)=R\) (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 \(r(\theta)=R\) (a constant), and so \(u=1 / R\). The polar equation of a circle passing through the origin is \(r=2 a \cos \theta\), 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, \(\overrightarrow{\mathbf{F}}^{a i r}=-b v^{2} \hat{\mathbf{v}}\), where \(\hat{\mathbf{v}}\) is a unit vector in th...The difference between the two paths is due to air resistance acting on the object, \(\overrightarrow{\mathbf{F}}^{a i r}=-b v^{2} \hat{\mathbf{v}}\), where \(\hat{\mathbf{v}}\) is a unit vector in the direction of the velocity. (For the orbits shown in Figure \(5.1, b=0.01 \mathrm{N} \cdot \mathrm{s}^{2} \cdot \mathrm{m}^{-2}\), \(\left|\overrightarrow{\mathbf{v}}_{0}\right|=30.0 \mathrm{m} \cdot \mathrm{s}\), the initial launch angle with respect to the horizontal \(\theta_{0}=21^{\circ}\) an…
