Question

The friction coefficient between a road and the type of a vehicle is 4/3. Find the maximum incline the road may have so that once had brakes are applied and the wheel starts skidding, the vehicle going down at a speed of 36 km/hr is stopped within 5 m.

Answer 1

Given,
initial velocity of the vehicle, u = 36 km/h = 10 m/s
final velocity of the vehicle, v = 0
s = 5m, μ = `4/3`, g = 10 m/s²
Let the maximum angle of incline be θ.
Using the equation of motion
`a=(v^2-u^2)/(2s)=(0-10)/(2xx5)`
= -10 m/s²
From the free body diagram

R − mg cos θ = 0
⇒ R = mg cos θ                     (1)
Again,
ma + mg sin θ − μ R = 0
⇒ ma + mg sin θ − μmg cos θ = 0
⇒ a + g sin θ − μg cos θ = 0
`=>10+10sintheta-(4/3)xx10cos theta = 0`
⇒ 30 + 30 sin θ − 40 cos θ = 0
⇒ 3 + 3 sin θ − 4 cos θ = 0
⇒ 4 cos θ − 3 sin θ = 3
`=>4sqrt(1-sin^2theta)` = 3 + 3 sin θ
On squaring, we get
16 (1 − sin² θ) = 9 + 9 sin² θ + 18 sin θ
25 sin² θ + 18 sin θ − 7 = 0
`=> sintheta=(18+sqrt(18^2-4(25)(-7)))/(2x25)`

`=(-18+32)/50=14/50=0.28`            (Taking positive sign only)
⇒ θ = sin^-1 (0.28) = 16°

Therefore, the maximum incline of the road, θ = 16°.