To find the velocity $V_2$ of the roller-coaster car at the third hill (height = 9 m), we'll apply the principle of conservation of mechanical energy. Assuming no energy loss (e.g., due to friction), the total mechanical energy remains constant throughout the motion.

Step 1: Initial Potential Energy

At the top of the first hill, the roller-coaster car has maximum potential energy and no kinetic energy. The formula for gravitational potential energy ($PE$) is:

$PE = mgh$

where:

• $m = 200 \, \text{kg}$ (mass of the car),
• $g = 10 \, \text{m/s}^2$ (acceleration due to gravity),
• $h = 17 \, \text{m}$ (height of the first hill).

$PE_{\text{initial}} = 200 \times 10 \times 17 = 34,000 \, \text{J}$

Step 2: Potential Energy at the Third Hill

At the third hill (height = 9 m), the potential energy is:

$PE_{\text{third}} = mgh = 200 \times 10 \times 9 = 18,000 \, \text{J}$

Step 3: Kinetic Energy at the Third Hill

Since mechanical energy is conserved:

$KE_{\text{third}} + PE_{\text{third}} = PE_{\text{initial}}$

Let $KE_{\text{third}}$ be the kinetic energy at the third hill:

$KE_{\text{third}} = PE_{\text{initial}} - PE_{\text{third}}$ $KE_{\text{third}} = 34,000 - 18,000 = 16,000 \, \text{J}$

Step 4: Solving for Velocity ($V_2$)

The kinetic energy ($KE$) is also given by:

$KE = \frac{1}{2}mv^2$

Substituting the known values:

$16,000 = \frac{1}{2} \times 200 \times V_2^2$ $16,000 = 100 \times V_2^2$ $V_2^2 = 160$ $V_2 = \sqrt{160}$ $V_2 \approx 12.65 \, \text{m/s}$

Final Answer:

The velocity $V_2$ of the roller-coaster car at the third hill is approximately 12.65 m/s.

Would you like more details or have any questions?

Related Questions:

1. What would be the velocity of the car at the second hill (height = 16 m)?
2. How would the velocity change if friction was considered in the calculations?
3. What is the kinetic energy of the roller-coaster car at the bottom of the first hill?
4. If the height of the first hill was increased, how would it affect $V_2$?
5. How does the mass of the roller-coaster car influence the velocity at different points?

Tip:

Always remember that in a frictionless environment, the mechanical energy (potential + kinetic) of a system remains constant.