The wings provide a total aerodynamic downforce and drag that is related to the
ID: 1898928 • Letter: T
Question
The wings provide a total aerodynamic downforce and drag that is related to the velocity of the car of the form (forcen N, velocity in m/s); Following eqns refer to cars with wings.down=2.6v^2
FDrag=1.5v^2
Without the wings, the car only has an aerodynamic drag force of the form;
Fdrag=1.0v^2
Cars have a mass of 600kg
and a coefficient of statics friction of 1.5.
(a) Calculate how many meters it would take for each version of the car to stop from a speed of 300km/hr.
(b) Calculate the time taken to stop in each case.
Explanation / Answer
Weight of the car = 600*9.81 = 5886 N
300 km/hr = 300*1000/3600 m/s = 83.33 m/s
a) For car with wings:
Downward force = 2.6v^2 + 5886
Friction force = 1.5*(2.6v^2 + 5886) = 3.9v^2 + 8829
Total force opposing the forward motion = (3.9v^2 + 8829) + 1.5v^2 = 5.4v^2 + 8829
Force = ma = m dv/dt = m (dv/ds)(ds/dt) = mv dv/ds
ds = mv/(5.4v^2 + 8829) dv
Integrating both the sides from position 1 to 2, s2 - s1 = m/(5.4*2) ln [(5.4(v2)^2 + 8829)/(5.4(v1)^2 + 8829)]
Hence, distance travelled s = 600/10.8 ln 8829/(5.4*83.33^2 + 8829) = -92 m (or 92 m)
For car without wings:
Downward force = 5886
Friction force = 1.5*5886 = 8829
Total force opposing the forward motion = 8829 + 1.0v^2 = v^2 + 8829
Force = ma = m dv/dt = m (dv/ds)(ds/dt) = mv dv/ds
ds = mv/(v^2 + 8829) dv
Integrating both the sides from position 1 to 2, s2 - s1 = m/2 ln [((v2)^2 + 8829)/((v1)^2 + 8829)]
Hence, distance travelled s = 600/2 ln 8829/(83.33^2 + 8829) = -174 m (or 174 m)
b) For car with wings:
Force = ma = m dv/dt = 5.4v^2 + 8829
dt = m dv/(5.4v^2 + 8829)
Integrating both the sides from position 1 to 2, t2 - t1 = m/(5.4*8829) [atan(v2(5.4/8829)) - atan(v1(5.4/8829))]
Time taken = 600/(5.4*8829) [atan(0) - atan(83.33(5.4/8829))] =
For car without wings:
Force = ma = m dv/dt = v^2 + 8829
dt = m dv/(v^2 + 8829)
Integrating both the sides from position 1 to 2, t2 - t1 = m/8829 [atan(v2(1/8829)) - atan(v1(1/8829))]
Time taken = 600/8829 [atan(0) - atan(83.33(1/8829))] =
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