calculus antiderivative of the function. Find the position of the particle
A particle is moving with the given data:
$$a(t)=\cos{t}+\sin{t},\, s(0)=8,\, v(0)=5.$$
Find the position $s(t)$ of the particle.
I don't understand the problem. What do those symbols stand for?
$\endgroup$ 22 Answers
$\begingroup$If we assume $s(t)$ to be position function then, $a(t)=s''(t),~v(t)=s'(t)$ according to the definition. Here, we have $$s''(t)=\cos(t)+\sin(t)$$ so by solving this ODE, we get $v(t)=s'(t)=\sin(t)-\cos(t)+C_1$ and by taking an integration again we get: $$s(t)=-\cos(t)-\sin(t)+C_1t+C_2$$ in which $C_1,C_2$ are some constants. Now use the initial values you 're given to find proper constants $C_1,C_2$.
$\endgroup$ $\begingroup$The problem is asking you to find the position given that you know the acceleration at all times and the position & speed at a distinguished time $t=0$.
$a$ stands for acceleration.
$a(t)$ stands for the acceleration $t$ time units after the distinguished time denoted by $t=0$.
$\cos t$ is the cosine of $t$ --- $\sin t$ is the sine of $t$. Do you need more information on these?
$s$ stands for distance/displacement from a distinguished point $O$.
$s(0)$ stands for the distance/displacement from $O$ at the time $t=0$.
$v$ stands for the speed/velocity.
$v(0)$ stands for the speed/velocity at the time $t=0$.
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