Question

6. A motor car, of a total mass 1400 kg, running in top gear at 55 km/h, passes on to a rsing gradient of 1 in 20 at that speed. The road and other resistances may be taken as constant at 270 N/t. With condition unchanged, the speed falls uniformly to 40 km/h in a distance of 360 m. An intermediate gear is then engaged for which the speed ratio between the engine and the road wheels is 9:1 and the transmission efficiency is 85 per cent. The engine is then developing a torque of 85 Nm
Determine;
a. The tractive effort before and after the gear change, given that the road wheels are 750 mm diameter, [851 N; 1735 kN]
b. The time between the start of retardation and the recovery of the original speed of 55 km/h and [36 s]
c. The power being developed by the engine when the car is retarding and accelerating through 50 km/h, given that the transmission efficiency in top gear is 94 per cent.         [12.56 kW; 28.35 kW]

Answer 1

Initial velocity u = 55 km/h = 15.28 m/s

Final velocity v = 40 km/h = 11.11 m/s

Using v² = u² + 2as we get

11.11² = 15.28² + 2 * a * 360

Acceleration a = - 0.1527 m/s²

Grade = Tan = 1 / 20

Angle =  2.862 deg

Force due to gravity in the direction of the slope = mg * sin = 1400 * 9.81 * sin 2.862 = 685.84 N

By forces balance on the car: ma = T - mg sin - 270.....where T is the tractive effort

1400 * (- 0.1527) = T - 685.84 - 270

- 213.799 = T - 955.84

T = 742.04 N

After gear change, Torque at wheel = 85 * (9/1) * 0.85 = 650.25 Nm

Wheel radius = 750 / 2 = 375 mm = 0.375 m

Tractive effort T = 650.25 / 0.375 = 1734 N

b)

Time required to slow down to 40 km/h can be found by Using v = u + at we get

t = (11.11 - 15.28) / (- 0.1527)

t = 27.28 s

New acceleration after gear change is given by ma = 1734 - 955.84

Putting m = 1400 kg, we get a = 0.556 m/s²

Time required to accelerate from 40 km/h to 55 km/h can be found by (15.28 - 11.11) / 0.556 = 7.5 s

Total time = 27.28 + 7.5 = 34.78 s

c)

50 km/h = 13.89 m/s

While retartding, power at wheel = 742.04 * 13.89 = 10307 Watts = 10.3 kW

Power at engine = 10.3 / 0.94 = 10.96 kW

While accelerating, power at wheel = 1734 * 13.89 = 24085 Watts = 24.09 kW

Power at engine = 24.09 / 0.85 = 28.35 kW