A roller-coaster vehicle has a mass of 506 kg when fully loaded with passengers (see figure).

(a) If the vehicle has a speed of 22.0 m/s at point A, what is the force of the track on the vehicle at this point?

(b) What is the maximum speed the vehicle can have at point B in order for gravity to hold it on the track?

point A, r=10m and point B, r=15m

1.) ((m*v^2)/r1)+ma

To solve these questions, we need to apply Newton's second law of motion and consider the forces acting on the roller-coaster vehicle.

(a) To find the force of the track on the vehicle at point A, we need to consider the forces acting on the vehicle. At any point on the track, there are two main forces acting on the vehicle: the force of gravity (weight) and the force of the track pushing or pulling on the vehicle.

The force of gravity is given by the equation Fg = mg, where m is the mass of the vehicle (506 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).

The force of the track on the vehicle at point A can be calculated using Newton's second law of motion, which states that the net force on an object is equal to the mass of the object times its acceleration:

Fnet = ma

In this case, the net force is the sum of the force of the track (Ftrack) and the force of gravity (Fg):

Fnet = Ftrack + Fg

To find the force of the track, we rearrange the equation:

Ftrack = Fnet - Fg

At point A, the vehicle is in motion with a speed of 22.0 m/s. Assuming point A is on a level track, there is no vertical acceleration. Therefore, the net force is equal to zero:

Fnet = 0

Using this information, we can calculate the force of the track at point A as:

Ftrack = 0 - Fg
Ftrack = 0 - (506 kg × 9.8 m/s^2)

(b) To find the maximum speed the vehicle can have at point B in order for gravity to hold it on the track, we need to consider the forces acting on the vehicle and the conditions for circular motion.

At the top of the loop (point B), the net force on the vehicle must be inward, towards the center of the loop, to keep the vehicle on the track. This inward force is provided by the combination of the force of gravity and the force of the track.

The maximum speed the vehicle can have at point B occurs when the force of gravity just equals the necessary inward force to keep the vehicle on the track.

Using the same equation as in part (a), Fnet = Ftrack + Fg:

Ftrack + Fg = 0

At point B, the acceleration of the vehicle is directed towards the center of the loop. The centripetal acceleration (ac) can be calculated using the following equation:

ac = v^2 / r

Where:
v is the velocity of the vehicle at point B (maximum speed),
r is the radius of the loop.

To find the maximum speed, we need to find the radius of the loop. This information is not provided in the question, so we cannot calculate the maximum speed without more information.

In summary, to solve these questions:
(a) Calculate the force of the track by subtracting the force of gravity from the net force at point A.
(b) Find the maximum speed by equating the force of gravity with the necessary inward force and using the equation for centripetal acceleration. However, we need the radius of the loop to calculate the maximum speed.

I have no idea what Point A , B are.