Suppose the roller coaster in Fig. 6-41 (h1 = 38 m, h2 = 11 m, h3 = 25) passes point 1 with a speed of 1.60 m/s. If the average force of friction is equal to one sixth of its weight, with what speed will it reach point 2? The distance traveled is 45.0 m.

To determine the speed at which the roller coaster will reach point 2, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system, which consists of kinetic energy (KE) and potential energy (PE), remains constant as long as there are no external forces acting on the system.

First, let's calculate the initial mechanical energy of the roller coaster at point 1. The mechanical energy is given by the sum of the kinetic and potential energies:

E1 = KE1 + PE1

The kinetic energy at point 1 is given as 1/2 mv1^2, where m is the mass of the roller coaster and v1 is the speed at point 1. The potential energy is given by mgh, where h is the height of the coaster at each point.

E1 = 1/2 mv1^2 + mgh1 -----(1)

Next, let's calculate the final mechanical energy at point 2. The final kinetic energy is given as 1/2 mv2^2, where v2 is the speed at point 2. The final potential energy is given by mgh2.

E2 = 1/2 mv2^2 + mgh2 -----(2)

According to the principle of conservation of mechanical energy, E1 = E2. Therefore, we can equate equations (1) and (2) and solve for v2.

1/2 mv1^2 + mgh1 = 1/2 mv2^2 + mgh2

Rearranging the equation, we get:

1/2 mv1^2 - 1/2 mv2^2 = mgh2 - mgh1

Dividing the equation by m, we get:

1/2 v1^2 - 1/2 v2^2 = gh2 - gh1

Given the values for h1, h2, v1, and the acceleration due to gravity (g), we can solve for v2.

Now, let's plug in the values into the equation and solve for v2:

1/2 (1.60 m/s)^2 - 1/2 v2^2 = (9.8 m/s^2)(11 m - 25 m)

Simplifying the equation:

0.4 m^2/s^2 - 1/2 v2^2 = -137.2 m^2/s^2

Rearranging the equation to solve for v2:

1/2 v2^2 = 0.4 m^2/s^2 + 137.2 m^2/s^2

v2^2 = 2(0.4 m^2/s^2 + 137.2 m^2/s^2)

v2^2 = 2(137.6 m^2/s^2)

v2^2 = 275.2 m^2/s^2

Taking the square root of both sides to solve for v2:

v2 = sqrt(275.2 m^2/s^2)

v2 ≈ 16.6 m/s

Therefore, the roller coaster will reach point 2 with a speed of approximately 16.6 m/s.