A curve of radius 40 m is banked so that a 1070 kg car traveling at 60 km/h can round it even if the road is so icy that the coefficient of static friction is approximately zero. The acceleration of gravity is 9.81 m/s

a)Find the minimum speed at which a car can travel around this curve without skidding if the coefficient of static friction between the road and the tires is 0.7.
Answer in units of m/s

b)Find the maximum speed under the same
conditions. Answer in units of m/s.

To find the minimum and maximum speeds at which a car can travel around the curve without skidding, we need to consider the forces acting on the car.

Let's start with the minimum speed:

a) Minimum speed:

At the minimum speed, the frictional force between the tires and the road must be sufficient to provide the centripetal force required to keep the car moving in a circle.

Given:
Radius of the curve, r = 40 m
Mass of the car, m = 1070 kg
Coefficient of static friction, μ = 0.7
Acceleration due to gravity, g = 9.81 m/s²

The centripetal force required is given by:
Fc = mv²/r

The maximum static frictional force available is given by:
Ffriction = μmg

For the car not to skid (i.e., for the frictional force to be equal to the centripetal force), we can equate these two forces:
Fc = Ffriction

mv²/r = μmg

Simplifying and solving for v, we get:
v = √(μrg)

Substituting the given values:
v = √(0.7 * 9.81 * 40)

Calculating this out, we get:
v ≈ 18.07 m/s

Therefore, the minimum speed at which the car can travel around the curve without skidding is approximately 18.07 m/s.

b) Maximum speed:

At the maximum speed, the frictional force between the tires and the road is at its maximum value of μmg. In this case, the frictional force provides both the centripetal force and opposes the outward sliding force.

The net force acting on the car can be expressed as:
Fnet = Fcentripetal - Ffriction

At the maximum speed, the net force is zero (no skidding). Therefore:
Fcentripetal = Ffriction

mv²/r = μmg

Simplifying and solving for v, we get:
v = √(μrg)

Substituting the given values:
v = √(0.7 * 9.81 * 40)

Calculating this out, we get:
v ≈ 18.07 m/s

Therefore, the maximum speed at which the car can travel around the curve without skidding is approximately 18.07 m/s.

To solve these problems, we need to consider the forces acting on the car as it rounds the curved road.

Let's begin by breaking down the forces acting on the car. The forces are the weight (mg) and the normal force (N) acting vertically, and the frictional force (f) acting horizontally.

a) Minimum speed without skidding:
To prevent skidding, the frictional force (f) must be sufficient to provide the necessary centripetal force (mv²/r). The frictional force is given by the formula f = μN, where μ is the coefficient of static friction and N is the normal force.

In this case, since the coefficient of static friction is approximated to zero, there is no frictional force present. Therefore, the minimum speed at which the car can travel around the curve without skidding is zero.

b) Maximum speed without skidding:
To find the maximum speed without skidding, we need to determine the maximum static frictional force that can be provided. The maximum static frictional force (f_max) is given by the formula f_max = μ_maxN, where μ_max is the maximum coefficient of static friction.

Given the coefficient of static friction between the road and the tires (μ = 0.7), we can calculate the maximum static frictional force (f_max).

The normal force (N) can be determined from the gravitational force acting on the car:
N = mg.

Using the given values: mass (m) = 1070 kg, and acceleration due to gravity (g) = 9.81 m/s², we can calculate the normal force (N) as follows:
N = (mass) × (acceleration due to gravity) = (1070 kg) × (9.81 m/s²).

Once we have the normal force (N), we can calculate the maximum static frictional force (f_max) using the formula:
f_max = μ_maxN = (maximum coefficient of static friction) × (normal force) = (0.7) × (N).

The maximum static frictional force must equal the centripetal force to prevent skidding:
f_max = mv²/r.

Given the radius of the curve (r) = 40 m, we can now solve the equation to find the maximum speed (v_max):
μ_maxN = mv_max²/r.

Rearranging the equation, we have:
v_max² = (μ_maxN × r) / m.

Taking the square root of both sides, we find:
v_max = √((μ_maxN × r) / m).

Substituting the known values, we get:
v_max = √((0.7×N×r) / m).

Finally, calculate v_max using the values of N, r, and m obtained earlier.

Remember to plug in the given values for mass, radius, and acceleration due to gravity to get the final answer in units of m/s.

a) 13.456 m/s

b)?