A curve of radius 20 m is banked so that a

970 kg car traveling at 50 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^2
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 0.3.
Answer in units of m/s

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

Thanks in advance :)

To find the minimum speed at which a car can travel around the curve without skidding, we need to consider the forces acting on the car. The car experiences two main forces: the gravitational force pulling it downward and the frictional force of the tires against the road.

1. Minimum speed without skidding:
At minimum speed, the frictional force is maximum and equals the centripetal force required to keep the car moving in a circular path without skidding. The formula for the centripetal force is:
Fc = m * v^2 / r
where Fc is the centripetal force, m is the mass of the car, v is its speed, and r is the radius of the curve.

In this case, the coefficient of static friction is 0.3, so the maximum frictional force is given by:
Ff_max = μ * N
where μ is the coefficient of static friction and N is the normal force acting on the car. On a banked curve, the normal force is given by:
N = mg * cos(theta)
where m is the mass of the car, g is the acceleration due to gravity, and theta is the angle of the banking.

Since the car is not skidding, the force due to friction is equal to the component of the car's weight perpendicular to the road surface. This can be given as:
Ff_max = mg * sin(theta)

Setting the centripetal force equal to the maximum frictional force, we have:
m * v^2 / r = mg * sin(theta)

Simplifying the equation, we can solve for the minimum speed v:
v_min = sqrt(r * g * sin(theta))

Using the given values, we have:
r = 20 m (radius of the curve)
g = 9.81 m/s^2 (acceleration due to gravity)
theta = angle of banking (not given)

Since the angle of banking is not provided, we cannot calculate the specific minimum speed without knowing the value of theta.

2. Maximum speed without skidding:
To find the maximum speed, we take into account the fact that the coefficient of static friction between the tires and the road is 0.3. We can use the same equation as before, but this time the maximum frictional force is given by:
Ff_max = μ * N
Substituting the value for Ff_max, we have:
m * v_max^2 / r = μ * N

Using the given values, we can calculate the maximum speed v_max:
v_max = sqrt(μ * r * g)

Substituting μ = 0.3, r = 20 m, and g = 9.81 m/s^2, we get:
v_max = sqrt(0.3 * 20 * 9.81) ≈ 7.67 m/s

Therefore, the maximum speed at which the car can travel around the curve without skidding is approximately 7.67 m/s.