At what distance from the intersection does the driver start to brake when the traffic light is red, if the car is moving

At what distance from the intersection does the driver start to brake when the traffic light is red, if the car is moving up the highway with an angle of inclination α = 30o at a speed of 60 km / h? The coefficient of friction between the tires and the road is 0.1.

∠α = 30 °.

V = 60 km / h = 16.7 m / s.

g = 10 m / s2.

μ = 0.1.

V = 0 m / s.

S -?

During braking, the traction force of the car Fт = 0.

2 Newton’s law in vector form will have the form: m * a = m * g + N + Ftr, where m * g is the force of gravity, N is the reaction force of the road surface, Ftr is the friction force of the wheels on the road ..

ОХ: m * a = Ftr + m * g * sinα.

OU: 0 = – m * g * cosα + N.

a = (Ftr + m * g * sinα) / m.

N = m * g * cosα.

The friction force Ffr is expressed by the formula: Ffr = μ * N = μ * m * g * cosα.

a = (μ * m * g * cosα + m * g * sinα) / m = g * (μ * cosα + sinα).

With uniformly accelerated movement, the path of the car S is expressed by the formula: S = (V0 ^ 2 – V ^ 2) / 2 * a.

Since the car in front of the traffic light must stop V = 0 m / s, the formula will take the form: S = V0 ^ 2/2 * a = V0 ^ 2/2 * g * (μ * cosα + sinα).

S = (16.7 m / s) ^ 2/2 * 10 m / s2 * (0.1 * cos30 ° + sin30 °) = 23.8 m.

Answer: the driver must start braking at a distance of at least S = 23.8 m from the traffic light.