Question

Q5b, is there any way to solve it without using trial and error?

Answer 1

There are various successive approximations you can use to solve 2t - cos t = 0. I don't know if you'd call them "trial and error".

For instance, Newton's method, where from each guess t you get the next guess t\_new by

t\_new = t - \[ f(t) / f'(t) \]= t - \[ (2t - cos t) /(2 + sin t) \]

If you start with the guess t = 1.0, this converges really rapidly:

    t = 1.0
    t = 0.486288017
    t = 0.450418605
    t = 0.450183622
    t = 0.450183611

Answer 2

Differentiating the equation of motion for x and setting it to be greater than zero shows that the time must be smaller than half a second.

The question comes down to finding the intercept between the functions y=2x and y=cos(x). As mentioned in previous answers Newton's method is a reliable way of calculating this.

As a physicist I would truncate the Taylor expansion of cos x and then solve a horrible polynomial on the computer to approximate the answer which is a little more than 0.45