The friction coefficient between a road and the tyre of a vehicle is 4/3. Find the maximum incline the road may have so that once hard brakes are applied and the wheel starts skidding, the vehicle going down at a speed of 36 km/hr is stopped within 5 m.
Let the maximum angle of incline be θ.
Initial velocity of the vehicle, u = 36 km/h = 10 m/s
Final velocity of the vehicle, v = 0
distance travel s = 5 m,
μ=4/3, g = 10 m/s2
Using the equation of motion
a = -10 m/s2
From the free body diagram
R = mg cosθ (1)
Again,
ma + mg sinθ = μR
⇒ ma + mg sinθ = μmg cosθ
⇒ a + g sinθ = μg cosθ
⇒ 10 +10 sinθ = 4/3×10 cosθ
⇒ 30 + 30 sinθ = 40 cos θ
⇒ 3 + 3 sinθ - 4 cos θ = 0
⇒ 4 cosθ − 3 sin θ = 3
⇒4(1-sin2 θ)1/2=3+3 sinθ
On squaring both sides, we get
16 (1 − sin2 θ) = 9 + 9 sin2 θ + 18 sinθ
25 sin2 θ + 18 sinθ − 7 = 0
⇒sinθ=0.28
⇒ θ = 16°
Therefore, the maximum incline of the road, θ = 16°