Question

Suppose an object moves in an elliptical orbit parametrized by x = 4 cos t and y = 6 sin t, where t ≥ 0. use (4) and (5) in section 6.8 in order to find a formula for the velocity v(t) of the object at a given time t. then determine the value of the velocity at the indicated times:

Answer 1

As we know that position is given as

$x = 4 cost , y = 6 sint$

now in order to find the velocity

$\frac{dx}{dt} = v_x$

so here we will have

$v_x = \frac{d}{dt} 4cost = -4sint$

similarly in y direction

$v_y = \frac{dy}{dt}$

$v_y = \frac{d}{dt}(6sint) = 6 cost$

now the net velocity is given as

$v = \sqrt{v_x^2 + v_y^2}$

$v = \sqrt{(-4sint)^2+(6cost)^2}$

$v = \sqrt{16sin^2t + 36 cos^2t}$

so above is the velocity as a function of time