Suppose we are given an arbitrary form for the acceleration a(t) of a particle that starts from rest at r(t_i)=0. What would be the derivation of its' position at r(t_f):
a)(t_f - t_i)⌠t_f a(t)dt
⌡t_i
b) v(t_f)+v(t_i)/2 ((t_f-t_i))
c)⌠t_f dt ⌠t dt'a(t') ???????
⌡t_i ⌡t_i
To derive the position at a given time, t_f, when we are given the acceleration function, a(t), and the initial position, r(t_i) = 0, we can use the following steps:
Step 1: Integrate the acceleration to find the velocity function.
Since acceleration is the derivative of velocity, we integrate the acceleration function, a(t), with respect to time, t, to obtain the velocity function, v(t).
This can be represented by the integral:
v(t) = ∫[t_i, t_f] a(t) dt
Step 2: Integrate the velocity function to find the position function.
Since velocity is the derivative of position, we integrate the velocity function, v(t), with respect to time, t, to obtain the position function, r(t).
This can be represented by the integral:
r(t) = ∫[t_i, t_f] v(t) dt
Step 3: Evaluate the position function at t = t_f to get the position at t_f.
Substitute t = t_f into the position function obtained in step 2 to compute the position value at time t_f.
This can be represented as:
r(t_f) = ∫[t_i, t_f] v(t) dt
Now let's compare the given options to see which one corresponds to the derived position formula:
a) (t_f - t_i)⌠t_f a(t)dt ⌡t_i
This option involves the integral of acceleration, but it is missing the second integration step to find the position.
b) v(t_f) + v(t_i) / 2 * (t_f - t_i)
This option includes the velocities at t_f and t_i and the time interval (t_f - t_i), but it does not involve integration.
c) ⌠t_f dt ⌠t dt'a(t')
This option includes multiple integrals, but it is missing the integration of velocity which is necessary to find the position.
From the given options, none of them corresponds to the correct formula for deriving the position at t_f.