Revision:Kinematics - The Student Room
The Student Room

Revision:Kinematics

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Kinematics


These notes are based on the requirements of the M3 A Level mathematics module.


1.1 - Acceleration depending on time (t)

\displaystyle a = \frac{dv}{dt}

\displaystyle v = \frac{dx}{dt}

\displaystyle a = \frac{d^2x}{dt^2}


\displaystyle a = f(t)

\displaystyle v = \int f(t) dt + c


\displaystyle v = g(t)

\displaystyle x = \int g(t) dt + k


1.2 - Acceleration depending on displacement (x)

\displaystyle a = F(x)

\displaystyle \frac{dv}{dt} = F(x)


By the chain rule,

\displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}


Therefore;

\displaystyle a  = \frac{dv}{dt} = \frac{dv}{dx} \times \frac{dx}{dt}


Since, \displaystyle \frac{dx}{dt} = v

\displaystyle a = v.\frac{dv}{dx}


If \displaystyle y = v^2 where \displaystyle v = f(x)

\displaystyle \frac{dy}{dx} = \frac{dy}{dv} \times \frac{dv}{dx} = 2v.\frac{dv}{dx}


OR rewrite \displaystyle y = v^2 as \displaystyle y = v.v

\displaystyle \frac{dy}{dx} = v.\frac{dv}{dx} + v.\frac{dv}{dx} = 2v.\frac{dv}{dx} (by the product rule)


Hence, \displaystyle \frac{d(v^2)}{dx} = 2v.\frac{dv}{dx}


The acceleration, therefore, may also be written;

\displaystyle a = \frac{d(\frac{1}{2}v^2)}{dx}


If \displaystyle v = f(x) = \frac{dx}{dt}

\displaystyle \frac{1}{v} = \frac{1}{f(x)} = \frac{1}{\frac{dx}{dt}} = \frac{dt}{dx}

\displaystyle \int \frac{1}{v} dx = \int 1 dt = t


Comments

Originally written by Widowmaker on TSR forums.

Discussions Toggle
LNAT results thread 2012
started by: christinasrose
forum: Law
replies: 530
last post: 3 Minutes Ago
What are your thoughts on this defence of bestiality
started by: Meus
forum: Society
replies: 49
last post: 3 Minutes Ago
The Persian Society (UKL Iranian Society)
started by: petzneo
forum: Foreign Languages
replies: 7300
last post: 4 Minutes Ago
Is it weird that i haven't worn underwear in about 7 years...
started by: The Entity
forum: Advice on Everyday Issues
replies: 39
last post: 5 Minutes Ago
University of Kent Applicants 2012
started by: Onique
forum: Kent Unis
replies: 1031
last post: 6 Minutes Ago
Vegetarianism/veganism and ' animals eating animals...'
started by: grey_chair
forum: Philosophy
replies: 91
last post: 7 Minutes Ago
What Book Are You Reading Now? Mk II
started by: Phalanges
forum: Books, Literature & Poetry
replies: 1772
last post: 9 Minutes Ago
Dentistry Graduate Entry 2012
started by: cptKernow
forum: Dentistry
replies: 648
last post: 9 Minutes Ago
Has anyone 'converted' to multiculturalism
started by: Thatcuber
forum: Society
replies: 178
last post: 14 Minutes Ago
Having none to talk to because you fallen out with your friends and family.
started by: da_comeback
forum: Friends, Family and Work
replies: 6
last post: 14 Minutes Ago
If you could star in any TV show which would it be?
started by: lolalola
forum: TV
replies: 24
last post: 14 Minutes Ago
King's College London - 4 year (GEP) Denistry 2012
started by: animal'11
forum: Dentistry
replies: 10
last post: 15 Minutes Ago
Breaking Bad Season 4
started by: freefrag
forum: TV
replies: 319
last post: 18 Minutes Ago
Deferring masters?
started by: turkishdisco
forum: Postgraduate
replies: 5
last post: 19 Minutes Ago
Trans-sexualism --> Trans-speciesism
started by: ollyb96
forum: Society
replies: 233
last post: 19 Minutes Ago
Can you study dentistry graduate with a dental hygiene & therapist degree???
started by: iJamal
forum: Dentistry
replies: 1
last post: 19 Minutes Ago
Argentina to complain to US about 'militarization' of the South Atlantic
started by: ras90
forum: International
replies: 5
last post: 19 Minutes Ago
Cambridge graduate applicants 2011/12
started by: Craghyrax
forum: Postgraduate
replies: 2211
last post: 23 Minutes Ago
Imperial College Medicine Applicants 2012
started by: oHellno
forum: Medical Schools
replies: 1977
last post: 23 Minutes Ago
How often do you 'go out'?
started by: Anonymous
forum: Friends, Family and Work
replies: 61
last post: 28 Minutes Ago
Article Updates Toggle
Contact Us | Site Rules | Staying Safe on TSR | Advertising | Staff Blog | Essays & Coursework | Terms & Conditions | Top
Customise your TSR | Life Advice | Hobbies and Interests | Debate and Current Affairs | Study Help | University and University courses
Universities and HE Colleges | Careers, Employment and Gap Years | General Discussion

Customise your TSR