To find the speed of the roller coaster at point 2, we will use energy conservation principles, accounting for the work done by friction.

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Step 1: Define the energy at point 1

The roller coaster's total energy at point 1 consists of its gravitational potential energy:

$E_1 = m \cdot g \cdot h_1$ where:

• $m$ = mass of the roller coaster
• $g = 9.81 \, \text{m/s}^2$ (acceleration due to gravity)
• $h_1 = 32 \, \text{m}$

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Step 2: Define the energy at point 2

At point 2, the roller coaster will have gravitational potential energy and kinetic energy:

$E_2 = m \cdot g \cdot h_2 + \frac{1}{2} m v^2$ where:

• $h_2 = 14 \, \text{m}$
• $v$ = speed at point 2 (what we are solving for)

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Step 3: Account for energy lost due to friction

The frictional force is given as $F_f = 0.25 \cdot m \cdot g$, and the work done by friction ($W_f$) over the horizontal distance is:

$W_f = F_f \cdot d = (0.25 \cdot m \cdot g) \cdot 50.0 \, \text{m}$

Substitute $g = 9.81 \, \text{m/s}^2$:

$W_f = 0.25 \cdot m \cdot 9.81 \cdot 50.0 = 122.625 \cdot m \, \text{J}$

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Step 4: Apply energy conservation

The total energy at point 1 minus the work done by friction equals the total energy at point 2:

$E_1 - W_f = E_2$

Substitute expressions for $E_1$ and $E_2$:

$m \cdot g \cdot h_1 - W_f = m \cdot g \cdot h_2 + \frac{1}{2} m v^2$

Cancel $m$ from all terms:

$g \cdot h_1 - 122.625 = g \cdot h_2 + \frac{1}{2} v^2$

Substitute known values ($g = 9.81 \, \text{m/s}^2, h_1 = 32 \, \text{m}, h_2 = 14 \, \text{m}$):

$9.81 \cdot 32 - 122.625 = 9.81 \cdot 14 + \frac{1}{2} v^2$

Calculate each term:

$313.92 - 122.625 = 137.34 + \frac{1}{2} v^2$

$191.295 = 137.34 + \frac{1}{2} v^2$

$53.955 = \frac{1}{2} v^2$

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Step 5: Solve for $v$

Multiply through by 2:

$v^2 = 107.91$

Take the square root:

$v = \sqrt{107.91} = 10.4 \, \text{m/s}$

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Final Answer:

$\boxed{10.4 \, \text{m/s}}$

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Let me know if you want more details or clarification! Here are 5 follow-up questions:

1. How is friction modeled in this problem?
2. How would the result change if there was no friction?
3. What is the physical meaning of energy conservation in this context?
4. How does the height difference affect the speed calculation?
5. Could this setup be analyzed using a different approach, such as forces?

Tip: Always check if frictional forces are significant, as they impact energy transformations!