Suppose that r is the position vector of a particle moving

Problem 1E In Exercises 1–5, find the velocity and acceleration vectors in terms of

In Exercises 1-5, find the velocity and acceleration vectors in terms of \(\mathrm{u}_{\mathrm{r}}\) and \(\mathrm{u}_{\theta}\). \(r=a \sin 2 \theta \quad \text { and } \quad \frac{d \theta}{d t}=\ 2 t\) Equation Transcription: Text Transcription: u_r u_theta r = a sin 2 theta and d theta/dt = 2t

In Exercises , find the velocity and acceleration vectors in terms of and . \(r=e^{a \theta} \text { and } \quad \frac{d \theta}{d t}=2\) Equation Transcription: Text Transcription: u_r u_theta r=e^a theta and d theta/dt =2

Problem 4E In Exercises 1–5, find the velocity and acceleration vectors in terms of

In Exercises 1-5, find the velocity and acceleration vectors in terms of \(\mathrm{u}_{\mathrm{r}}\) and \(\mathrm{u}_{\theta}\). \(r=2 \cos 4 t \quad \text { and } \quad \theta=2 t\) Equation Transcription: Text Transcription: u_r u_theta r = 2 cos 4t and theta = 2t

Problem 6E Type of orbit For what values of v0 in Equation (5) is the orbit in Equation (6) a circle? An ellipse? A parabola? A hyperbola?

Problem 7E Circular orbits Show that a planet in a circular orbit moves with a constant speed. (Hint: This is a consequence of one of Kepler’s laws.)

Problem 8E Suppose that r is the position vector of a particle moving along a plane curve and dA dt is the rate at which the vector sweeps out area. Without introducing coordinates, and assuming the necessary derivatives exist, give a geometric argument based on increments and limits for the validity of the equation

Kepler’s third law Complete the derivation of Kepler’s third law (the part following Equation \((10)\)). Equation Transcription: Text Transcription: (10)

Problem 10E Find the length of the major axis of Earth’s orbit using Kepler’s third law and the fact that Earth’s orbital period is 365.256 days.