Final answer:
The magnitude of the centripetal acceleration is 1.25 m/s² and the car needs to travel at a speed of 25.0 m/s.
Step-by-step explanation:
In this scenario, there is no friction, which implies that Fn must provide the necessary centripetal force to keep the car moving in a circle. Since we are told to ignore vertical motion, gravity and normal force components perpendicular to the horizontal plane cancel out, and do not affect the centripetal motion directly. Therefore, the mass of the car (m) cancels out when equating the centripetal force (m × ac) with the horizontal component of the normal force, and we are left with ac = v2/r as our working formula. Substituting our given values for v (speed of the car) and r (radius of the curve), we can solve for the centripetal acceleration and the required speed.
The magnitude of the centripetal acceleration can be found using the formula:
ac = v^2 / r
where v is the velocity and r is the radius of the curve. Using the given values, the centripetal acceleration is:
ac = (25.0 m/s)^2 / 500 m
= 1.25 m/s^2
The car needs to travel at a speed of 25.0 m/s to maintain this centripetal acceleration.