A car, having a mass of 1.2 x 10^3kg rounds a curve which is banked at an angle of 18 degrees to the horizontal. The radius of the curve is 130m. Find the speed of the car if the frictional force between the tires and the road is 3.5 x 10^3N down the banked curve.

To find the speed of the car, we can use the concept of centripetal force and the relationship between the gravitational force, normal force, and frictional force on a banked curve.

First, let's analyze the forces acting on the car:

1. Gravitational force (mg): This force acts vertically downwards and has a magnitude of mg, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s²).

2. Normal force (N): This force acts perpendicular to the surface of the curve and balances the vertical component of the gravitational force. It can be calculated as N = mg * cos(θ), where θ is the angle of the banked curve (18 degrees in this case).

3. Frictional force (f): This force acts parallel to the surface of the curve. It opposes the motion of the car and helps provide the necessary centripetal force to keep the car moving in a circular path. Given that the frictional force is 3.5 x 10³ N down the banked curve, we can take its magnitude as f = 3.5 x 10³ N.

4. Centripetal force (Fc): This force acts towards the center of the circular path and is responsible for keeping the car moving in a curve without slipping. The centripetal force can be calculated as Fc = (mv²) / r, where v is the speed of the car and r is the radius of the curve.

Now, let's find the speed of the car using the information given:

Step 1: Calculate the normal force (N):
N = mg * cos(θ)
= (1.2 x 10³ kg)(9.8 m/s²)(cos 18°)

Step 2: Calculate the centripetal force (Fc):
Fc = (mv²) / r
= (1.2 x 10³ kg)(v²) / 130 m

Step 3: Equate the centripetal force to the frictional force:
Fc = f

(1.2 x 10³ kg)(v²) / 130 m = 3.5 x 10³ N

Step 4: Rearrange the equation and solve for v:
v² = (3.5 x 10³ N)(130 m) / (1.2 x 10³ kg)
v² = 3.8075 x 10⁴ m²/s²

v = √(3.8075 x 10⁴ )
v ≈ 195.123 m/s

Therefore, the speed of the car is approximately 195.123 m/s.