Answer to Question #103188 in Classical Mechanics for Martin

Question #103188
(7) As a car of mass m kg drives up a slope at an angle α to the horizontal it experiences a
constant resistive force FN and a driving force D N.What can be deduced about the work
done by D as the car moves a distance d metres uphill if:
(i) the car moves at constant speed?
(ii) the car slows down?
(iii) the car gains speed?
The initial and final speeds of the car are denoted by um/s and vm/s respectively.
(iv) Write v
2
in terms of the other variables.
1
Expert's answer
2020-02-17T09:25:12-0500

(i) As the car moves at a constant speed "v", its work increases linearly according to the expression


"W_\\text{i}=Dv\\text{ cos}\\alpha."

(ii) Since the speed decreases from "v", the work decreases as compared to previous example:


"W_\\text{ii}<W_\\text{i}."

(iii) When the car gains speed starting from "v", the work increases compared to the first example:


"W_\\text{iii}>W_\\text{i}."

2. Newton's second law states that acceleration is proportional to the net force:


"a=\\frac{F_\\text{net}}{m}=\\frac{D-F}{m}."

On the other hand, the acceleration can be expressed in terms of speeds v and u and distance d:

"a=\\frac{v^2-u^2}{2d}."

Hence:


"v=2ad+u^2=u^2+\\frac{2d}{m}(D-F)."


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