Answer to Question #103188 in Classical Mechanics for Martin

Question

(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.

Answer 1

(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:

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