To determine the magnitude of the retarding force acting on the roller coaster, we can use the work-energy principle. The work done by the retarding force is equal to the change in the kinetic energy of the roller coaster.

The formula for kinetic energy is:

$KE = \frac{1}{2} m v^2$

where:

• $m$ is the mass of the roller coaster (4768 kg),
• $v$ is the velocity (initial and final).

Step 1: Calculate the initial and final kinetic energies.

• Initial velocity, $v_i = 17.1 \, \text{m/s}$
• Final velocity, $v_f = 2.20 \, \text{m/s}$

The initial kinetic energy is:

$KE_i = \frac{1}{2} m v_i^2 = \frac{1}{2} (4768) (17.1^2)$

The final kinetic energy is:

$KE_f = \frac{1}{2} m v_f^2 = \frac{1}{2} (4768) (2.2^2)$

Step 2: Find the change in kinetic energy.

$\Delta KE = KE_f - KE_i$

Step 3: Use the work-energy theorem to find the retarding force.

The work done by the retarding force $F$ is:

$W = F \cdot d = \Delta KE$

where $d = 13.6 \, \text{m}$ is the distance over which the roller coaster is decelerated.

Thus, the force is:

$F = \frac{\Delta KE}{d}$

Let’s calculate that now.The magnitude of the retarding force acting on the roller coaster is approximately $50,409 \, \text{N}$. Since it's a retarding force, the negative sign indicates that the force is acting in the direction opposite to the motion.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the work-energy theorem apply to real-life scenarios beyond roller coasters?
2. How would the force change if the stopping distance were increased?
3. What factors, besides force, affect the deceleration of objects in motion?
4. How can you calculate the power exerted by the retarding force during deceleration?
5. What role does friction play in the deceleration of a moving object?

Tip: The work-energy theorem can be a powerful tool to solve motion problems when forces aren't directly provided.