A roller coaster heads into a circular loop with radius 5m. At what minimum speed must the coaster go at the top of the loop to be sure it stays on the track?

I don't really understand what I am supposed to be solving for.

I am not certain about that at all.

Gravity is pushing the coaster down. to keep in on the loop, gravity must be greater than a force needed to keep the coaster in a circular path, if the force of gravity is not greater, the coaster will fly off the track.
The speed of "decision" is when gravity equals the force needed to keep the coaster on the track.
mg=mv^2/r
v=sqrt(rg)=sqrt(5*9.8) about 7m/s. This is the minimum speed to keep in on the track. Above this, the coaster flys off.
Gravity keeps the train on. If the coaster is faster, something else (magnets, or clamps, or ???) has to be attached to keep it on the track, to supply the inward (downward) force on the coaster to accelerate it downward.

gravity and the curved track are supplying the centripetal force

if the required force is less than gravity (too slow in the loop), the coaster will not stay on the track

Depends on if the train is under the loop looking up or over the loop looking down.

Since it says "minimum" speed I suspect we are under the track and need big speed to keep from dropping
mv^2/R > mg
or centripetal acceleration greater than g

Thank you!

To solve this problem, we need to find the minimum speed at the top of the loop necessary for the roller coaster to stay on the track. In order to stay on the track, the centripetal force experienced by the roller coaster must be greater than or equal to the gravitational force pulling it downwards.

Let's break down the problem:

1. Identify the forces: The only two significant forces acting on the roller coaster are the gravitational force (mg) and the centripetal force (mv²/r), where m is the mass of the roller coaster, g is the gravitational acceleration (approximately 9.8 m/s²), v is the velocity of the roller coaster, and r is the radius of the loop.

2. Equate the forces: At the top of the loop, the gravitational force and the centripetal force should be equal or the centripetal force should be greater than the gravitational force to ensure the roller coaster stays on the track. This can be expressed as:

mv²/r ≥ mg

3. Solve for the minimum speed: Rearrange the equation to solve for the minimum speed:

v² ≥ rg

v ≥ sqrt(rg)

where sqrt represents the square root.

4. Substitute the values: Plug in the given radius into the equation:

v ≥ sqrt(5 * 9.8)

Calculate the value inside the square root:

v ≥ sqrt(49)

Simplify:

v ≥ 7 m/s

Therefore, the minimum speed the coaster must go at the top of the loop to ensure it stays on the track is 7 m/s.

Remember, this solution assumes no friction or air resistance. Additionally, keep in mind that this is just the minimum speed required for the coaster to stay on the track, and going faster will provide additional stability.