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

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
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

0 like 0 dislike