A 1600 kg car, when on a rectilinear and horizontal stretch of road and when the tires do not slip, can go from rest to 100 km/h in 5.5 s. Assuming that the car travels on a straight stretch of road with a 40% slope and that no slip occurs, determine how long it would take to attain a speed of 100

Question

A 1600 kg car, when on a rectilinear and horizontal stretch of road and when the tires do not slip, can go from rest to 100 km/h in 5.5 s. Assuming that the car travels on a straight stretch of road with a 40% slope and that no slip occurs, determine how long it would take to attain a speed of 100 km/h if the car were propelled by the same maximum average force that can be generated on a horizontal road.

Accepted Answer

We start the solution of this problem by determining the average force that acts on the car when it travels on a horizontal surface going from zero to $100 km / s in 5.5 s. Let the subscripts 1 and 2 denote the time instants when the car starts from rest and reaches the speed of 100 km / h, respectively. Then, letting [latex]v_{h 1} ext { and } v_{h 2}[/latex] be the horizontal components of velocity corresponding to the the time instants [latex]t_1 ext { and } t_2[/latex] respectively, using the relation between the notion of average force and the change in momentum (over a given time interval), we have

[latex]F_{ ext {avg }}=\frac{m v_{h 2}-m v_{h 1}}{t_2-t_1} \Rightarrow F_{ avg }=8081 \ N,[/latex] (1)

where we have used the following numerical data: $m=1600 kg , [latex]v_{h 2}=(100 \ km / h )(3600 \ s / hr )^{-1}[/latex] = 27.78 m / s , [latex]v_{h 1}=0, ext { and } t_2-t_1=5.50 \ s[/latex].

Now, we turn the analysis of the motion of the car along the incline. Referring to the FBD shown on the right, as indicated by the problem statement, we assume that the car is subject to a force equal to [latex]f_{ ext {avg }}[/latex]acting in the direction of motion. Hence, applying Newton's second law in the x direction, we have

[latex]\sum{F_x} :-mg \sin heta +F_{avg}=ma_x ,\Rightarrow a_x =\frac{F_{avg}}{m}- g \sin heta[/latex]

Question : A 1600 kg car, when on a rectilinear and horizontal stretch ...

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