00:01
Hello everyone, so we will solve this question now.
00:02
According to the question, let us write the given part first of all.
00:06
So it is given that acceleration at any time t, this vector is given as cos 3t, sin 3t, 0.
00:17
Okay.
00:18
And also velocity at t is equal to zero, it is given as 0, 1, 6.
00:27
And position vector at t is equal to zero, it is 2, 2, 2.
00:32
Okay.
00:33
So in the a part, we have to find the velocity at any time t.
00:37
Okay.
00:38
So acceleration can be written as we know that acceleration, it can be written as dv vector upon dt.
00:45
Okay.
00:45
And its value is cos 3t, sin 3t, 0.
00:52
Okay.
00:53
So if we want to write here, what value we will get from here, the dv vector value will be equal to cos 3t, i cap plus, plus here, sin 3t, j cap plus 0 k cap, or let us not write it.
01:13
Okay, this value.
01:14
And we are doing here, we will multiply by dt.
01:19
So if we integrate both the sides, the limits will be when time t is equal to zero, let us say the velocity is v vector zero.
01:28
And when the time is t, let us say the velocity is v vector t.
01:32
Okay.
01:33
So here, if we do integration, it will be v vector and upper limit minus lower limit, that means v vector t minus v vector zero.
01:41
Okay.
01:42
And here integration of cos 3t will be sin 3t, sin 3t upon 3, i cap plus integration of sin 3t will be minus of cos 3t upon 3, j cap, this value.
02:01
And we will put limits from zero to t.
02:03
Now we will do upper limit minus lower limit from here.
02:07
So v vector t minus v vector zero, this value will be equal to upper limit that means sin 3t upon 3, i cap, sin 3t upon 3, i cap, minus cos 3t upon 3, j cap.
02:27
Okay.
02:28
And minus of, if we put zero here, then first value, it will be zero and minus and minus it will be plus.
02:36
Okay.
02:36
And cos zero value is always one.
02:39
Okay.
02:40
So it will be one by three j cap here.
02:42
So one by three j cap.
02:45
So what we can write from here sin 3t upon 3, i cap.
02:51
And if we take plus here, okay, so it will be one by three minus cos 3t by three j cap, this value...