An object's two dimensional velocity is given by v(t) = ( sqrt(t-2)-t , t^2). What is the object's rate and direction of acceleration at t=3 ?

Sep 26, 2016

The vector is $a \left(t\right) = \left(\left(\frac{1}{2}\right) {\left(t - 2\right)}^{- \frac{1}{2}} - 1 , 2 t\right)$ Substitute 3 for t, and then compute the magnitude and direction as you would any vector.

Explanation:

$a \left(3\right) = \left(\left(\frac{1}{2}\right) {\left(3 - 2\right)}^{- \frac{1}{2}} - 1 , 2 \left(3\right)\right)$

$a \left(3\right) = \left(- \frac{1}{2} , 6\right)$

The rate of acceleration is the magnitude:

$| a \left(3\right) | = \sqrt{{\left(- \frac{1}{2}\right)}^{2} + {6}^{2}}$

$a \approx 6.02 \frac{u n i t s}{s} ^ 2$

$\theta = {\tan}^{-} 1 \left(\frac{6}{- \frac{1}{2}}\right)$

theta ~~ -85.2°
{"ft_lang_label":"__label__en","ft_lang_prob":0.7778087,"math_prob":1.0000085,"size":373,"snap":"2019-43-2019-47","text_gpt3_token_len":96,"char_repetition_ratio":0.10298103,"word_repetition_ratio":0.0,"special_character_ratio":0.26541555,"punctuation_ratio":0.097222224,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999838,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-22T13:40:54Z\",\"WARC-Record-ID\":\"<urn:uuid:94934738-9c45-49d7-8dff-3eb994185c89>\",\"Content-Length\":\"33988\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a025580c-ace2-4f0c-a587-e61f11644817>\",\"WARC-Concurrent-To\":\"<urn:uuid:075f5688-63d9-4599-bf3d-29595603298a>\",\"WARC-IP-Address\":\"54.221.217.175\",\"WARC-Target-URI\":\"https://socratic.org/questions/an-object-s-two-dimensional-velocity-is-given-by-v-t-sqrt-t-2-t-t-2-what-is-the-\",\"WARC-Payload-Digest\":\"sha1:C4MDGCB3PWWZEJQC3NIUNMUAZPF6ACFT\",\"WARC-Block-Digest\":\"sha1:5BCECFJI4ODZGFY437ASKPIKP7MBSRRY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987822098.86_warc_CC-MAIN-20191022132135-20191022155635-00153.warc.gz\"}"}