Suppose the roller coaster in the figure (h1 = 40 m, h2 = 14 m, h3 = 30) passes point A with a speed of 2.60 m/s. If the average force of friction is equal to one fifth of its weight, with what speed will it reach point B? The distance traveled is 35.0 m.

To find the speed at which the roller coaster will reach point B, we can apply the principle of conservation of mechanical energy. This principle states that the total mechanical energy of an object is conserved as long as there are no non-conservative forces acting on it, such as friction.

First, let's calculate the initial mechanical energy of the roller coaster at point A. The mechanical energy consists of two components: the potential energy due to the height and the kinetic energy due to the speed. The equation for mechanical energy is:

E = PE + KE

The potential energy at point A is given by the equation:

PE = m * g * h1

where m is the mass of the roller coaster, g is the acceleration due to gravity, and h1 is the height at point A. We can calculate this using the given information.

PE = m * 9.8 m/s^2 * 40 m

Next, the kinetic energy at point A is given by the equation:

KE = 0.5 * m * v^2

where m is the mass of the roller coaster and v is the speed at point A. We can calculate this using the given information.

KE = 0.5 * m * (2.60 m/s)^2

Now, let's find the final mechanical energy at point B. The mechanical energy at point B will consist of the potential energy due to the height and the kinetic energy due to the speed. The potential energy at point B is given by the equation:

PE = m * g * h2

where m is the mass of the roller coaster, g is the acceleration due to gravity, and h2 is the height at point B. We can calculate this using the given information.

PE = m * 9.8 m/s^2 * 14 m

Finally, the kinetic energy at point B can be calculated using the equation:

KE = 0.5 * m * v^2

where m is the mass of the roller coaster and v is the speed at point B. This is what we need to find.

Now, since we know that the mechanical energy is conserved, we can equate the initial mechanical energy (E at point A) to the final mechanical energy (E at point B):

PE + KE = PE + KE

(m * 9.8 m/s^2 * 40 m) + (0.5 * m * (2.60 m/s)^2) = (m * 9.8 m/s^2 * 14 m) + (0.5 * m * v^2)

Simplifying the equation:

400 m * g + 0.5 * m * (2.60 m/s)^2 = 140 m * g + 0.5 * m * v^2

Now, we can solve for v by rearranging the equation and isolating v:

v^2 = (400 m * g + 0.5 * m * (2.60 m/s)^2 - 140 m * g) / (0.5 * m)

v^2 = (400 * 9.8 m/s^2 - 140 * 9.8 m/s^2 + 0.5 * (2.60 m/s)^2) / 0.5

v = square root of [(400 * 9.8 m/s^2 - 140 * 9.8 m/s^2 + 0.5 * (2.60 m/s)^2) / 0.5]

Now, we can substitute the given values into the equation and calculate the result.