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Question Number 3463 by prakash jain last updated on 13/Dec/15
A car starts from rest and uniformly  accelerates for 1 km, travels with uniform  velocity for 98 km and brakes and stop  after travelling another 1 km.  Consider the above experiment being done  on 2 different roads 1st roads has higher  coefficient of friction for car for all types  than 2nd road.  Compare energy consumed in 2 cases, which one  consumes more energy and why?
$$\mathrm{A}\:\mathrm{car}\:\mathrm{starts}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{and}\:\mathrm{uniformly} \\ $$$$\mathrm{accelerates}\:\mathrm{for}\:\mathrm{1}\:\mathrm{km},\:\mathrm{travels}\:\mathrm{with}\:\mathrm{uniform} \\ $$$$\mathrm{velocity}\:\mathrm{for}\:\mathrm{98}\:\mathrm{km}\:\mathrm{and}\:\mathrm{brakes}\:\mathrm{and}\:\mathrm{stop} \\ $$$$\mathrm{after}\:\mathrm{travelling}\:\mathrm{another}\:\mathrm{1}\:\mathrm{km}. \\ $$$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{above}\:\mathrm{experiment}\:\mathrm{being}\:\mathrm{done} \\ $$$$\mathrm{on}\:\mathrm{2}\:\mathrm{different}\:\mathrm{roads}\:\mathrm{1st}\:\mathrm{roads}\:\mathrm{has}\:\mathrm{higher} \\ $$$$\mathrm{coefficient}\:\mathrm{of}\:\mathrm{friction}\:\mathrm{for}\:\mathrm{car}\:\mathrm{for}\:\mathrm{all}\:\mathrm{types} \\ $$$$\mathrm{than}\:\mathrm{2nd}\:\mathrm{road}. \\ $$$$\mathrm{Compare}\:\mathrm{energy}\:\mathrm{consumed}\:\mathrm{in}\:\mathrm{2}\:\mathrm{cases},\:\mathrm{which}\:\mathrm{one} \\ $$$$\mathrm{consumes}\:\mathrm{more}\:\mathrm{energy}\:\mathrm{and}\:\mathrm{why}? \\ $$
Commented by prakash jain last updated on 13/Dec/15
Same car is used on both tracks. All other  things equal only friction changes.
$$\mathrm{Same}\:\mathrm{car}\:\mathrm{is}\:\mathrm{used}\:\mathrm{on}\:\mathrm{both}\:\mathrm{tracks}.\:\mathrm{All}\:\mathrm{other} \\ $$$$\mathrm{things}\:\mathrm{equal}\:\mathrm{only}\:\mathrm{friction}\:\mathrm{changes}. \\ $$
Answered by Yozzii last updated on 13/Dec/15
The energy W expended by a body moving against  frictional force f is W=fd where d=distance  travelled for which f acts.  In situations of motion along a road  with a certain coefficient of friction μ,  f takes a maximum value of μR, where  R=normal reaction on body by the road.  ∴ W=μRd. If d and R are constant only,  ⇒ W∝μ.   R=mg by Newton′s 3rd Law of Motion  where m=mass of body, g=acceleration  due to gravity (dependent upon planet  where the experiment is held).    In both cases d=constant (100km) and any  valid comparison must have the   experiment performed with identical  cars on the same track, so that R is constant.  Since W∝μ, the track with higher μ  incurs higher energy consumption on  a given car.
$${The}\:{energy}\:{W}\:{expended}\:{by}\:{a}\:{body}\:{moving}\:{against} \\ $$$${frictional}\:{force}\:{f}\:{is}\:{W}={fd}\:{where}\:{d}={distance} \\ $$$${travelled}\:{for}\:{which}\:{f}\:{acts}. \\ $$$${In}\:{situations}\:{of}\:{motion}\:{along}\:{a}\:{road} \\ $$$${with}\:{a}\:{certain}\:{coefficient}\:{of}\:{friction}\:\mu, \\ $$$${f}\:{takes}\:{a}\:{maximum}\:{value}\:{of}\:\mu{R},\:{where} \\ $$$${R}={normal}\:{reaction}\:{on}\:{body}\:{by}\:{the}\:{road}. \\ $$$$\therefore\:{W}=\mu{Rd}.\:{If}\:{d}\:{and}\:{R}\:{are}\:{constant}\:{only}, \\ $$$$\Rightarrow\:{W}\propto\mu.\: \\ $$$${R}={mg}\:{by}\:{Newton}'{s}\:\mathrm{3}{rd}\:{Law}\:{of}\:{Motion} \\ $$$${where}\:{m}={mass}\:{of}\:{body},\:{g}={acceleration} \\ $$$${due}\:{to}\:{gravity}\:\left({dependent}\:{upon}\:{planet}\right. \\ $$$$\left.{where}\:{the}\:{experiment}\:{is}\:{held}\right). \\ $$$$ \\ $$$${In}\:{both}\:{cases}\:{d}={constant}\:\left(\mathrm{100}{km}\right)\:{and}\:{any} \\ $$$${valid}\:{comparison}\:{must}\:{have}\:{the}\: \\ $$$${experiment}\:{performed}\:{with}\:{identical} \\ $$$${cars}\:{on}\:{the}\:{same}\:{track},\:{so}\:{that}\:{R}\:{is}\:{constant}. \\ $$$${Since}\:{W}\propto\mu,\:{the}\:{track}\:{with}\:{higher}\:\mu \\ $$$${incurs}\:{higher}\:{energy}\:{consumption}\:{on} \\ $$$${a}\:{given}\:{car}.\: \\ $$$$ \\ $$

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