# A particle moves on a straight line with velocity function v(t)= sin((alpha)(t))cos((alpha)(t))^2 . Find its position function s(t) if s(0) = 2.

sciencesolve | Certified Educator

You need to integrate the given velocity function to find the position of particle such that:

`s(t) = int (sin alpha*t*cos alpha*t)^2 dt`

You need to use the half angle identities such that:

`sin^2 alpha*t = (1 - cos 2alpha*t)/2`

`cos^2 alpha*t = (1+ cos 2alpha*t)/2`

`s(t) = int (1 - cos 2alpha*t)/2*(1 + cos 2alpha*t)/2 dt`

`s(t) = (1/4)int (1 - cos 2alpha*t)(1 +cos 2alpha*t) dt`

You need to convert the integrand into a difference of squares such that:

`s(t) = (1/4)int (1 - cos^2 (2alpha*t)) dt`

Using the property of linearity such that:

`s(t) = (1/4)int dt- (1/4)int (cos^2 (2alpha*t)) dt`

`cos^2 (2alpha*t) = (1 + cos 4alpha*t)/2`

`s(t) = (1/4)int dt- (1/4)int(1 + cos 4alpha*t)/2 dt`

`s(t) = (1/4)int dt - (1/8)int (1 + cos 4alpha*t) dt`

`s(t) = (1/4)int dt - (1/8)int dt - (1/8)int (cos (4alpha*t)) dt`

`s(t) = (1/8)int dt - (1/8)int (cos (4alpha*t)) dt`

`s(t) = (1/8)(t - (sin (4alpha*t))/4) + c`

The problem provides the information that `s(0) = 2` , hence, you may evaluate the constant `c`  such that:

`s(0) = (1/8)(0 - (sin (0))/4) + c`

Since si`n 0 = 0 ` yields:

`s(0) = c => c = 2`

Hence, evaluating the position of the particle, under the given conditions, yields `s(t) = (1/8)(t - (sin (4alpha*t))/4) + 2` .

jeew-m | Certified Educator

V(t)

`= sinalphat*(cosalphat)^2`

Position function is given by the integration of vector function.

S(t)

`= int V(t)dt`

`= intsinalphat*(cosalphat)^2*dt`

Let

`x = cosalphat`

`(dx)/dt = -sinalphat*alpha`

`dx = -alpha*sinalphat*dt`

S(t)

`= intsinalphat*(cosalphat)^2*dt`

`= int -alpha*x^2dx`

`= -alpha int x^2dx`

`= -alpha(x^3/3)+C`

`= -alpha((cosalphat)^3/3)+C`

It is given that S(0) = 2

`2 = -alpha((cosalpha*0)^3/3)+C`

`2 = -alpha/3+C`

`C = 2+alpha/3`

`S(t) = -alpha((cosalphat)^3/3)+2+alpha/3`

So the position function is;

`S(t) = (alpha/3)(1-(cosalphat)^3)+2`
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.enotes.com/homework-help/particle-moves-straight-line-with-velocity-366037", "fetch_time": 1516555191000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2018-05/segments/1516084890795.64/warc/CC-MAIN-20180121155718-20180121175718-00152.warc.gz", "warc_record_offset": 926573071, "warc_record_length": 10165, "token_count": 794, "char_count": 2044, "score": 4.125, "int_score": 4, "crawl": "CC-MAIN-2018-05", "snapshot_type": "latest", "language": "en", "language_score": 0.6405727863311768, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00002-of-00064.parquet"}