How to find the force of a particle at $t=0$ which trajectory equation is $\sigma(t)=\cos(t)\hat{i}+\sin(2t)\hat{j}$?

Question

Problem:

A particle of 3 grams of mass follows the trajectory $\sigma(t)=\cos(t)\hat{i}+\sin(2t)\hat{j}$ (units in seconds and centimeters). Compute the force performed over the particle at $t=0$.

Solution:

The first thing I did was plot the parametric equation on Wolfram|Alpha (https://www.wolframalpha.com/input/?i=parametric+plot+(cos(t),+sin(2t))):

The section marked with the red figure is where the force must be found.

I know that Newton's law for force is given by $\vec{F}=ma(t)$. In this case $a=3gr$, but how do I find the expression $a(t)$?

Answer 1

As arctic tern said, acceleration is the second derivative of position.

Then $\vec{a}(t) = -$cos$(t) \hat{i} - 4$sin$(2t) \hat{j} $.

From there we have that $\vec{F}(t) = m\vec{a}(t) = -3$cos$(t) \hat{i} - 12$sin$(2t) \hat{j}$, so that $\vec{F}(0) = -3\hat{i} - 0\hat{j} = -3\hat{i}$.