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Find the velocity v2 for the roller-coaster car, use g=10 m/s2 the initial height is 17

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.

Find the velocity V2 for the roller-coaster car, use g=10 m/s² the initial height is 17 meters

Learn how to calculate the velocity of a 200 kg roller-coaster car at the third hill (9 m height) using conservation of mechanical energy. Includes step-by-step solution using g = 10 m/s² and key physics concepts such as potential and kinetic energy.

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