A roller coaster starts from rest at a height of 50 m. What is its speed at the lowest point? (g = 9.8 m/s²)

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Q: A roller coaster starts from rest at a height of 50 m. What is its speed at the lowest point? (g = 9.8 m/s²)

Solution: Using conservation of energy, mgh = 0.5mv². Thus, v = sqrt(2gh) = sqrt(2 * 9.8 * 50) = 31.3 m/s, approximately 30 m/s.

Steps: 11

Step 1: Understand that the roller coaster starts from a height of 50 meters and is initially at rest, meaning its initial speed is 0 m/s.
Step 2: Recognize that as the roller coaster descends, potential energy (due to its height) is converted into kinetic energy (due to its speed).
Step 3: Use the formula for potential energy: PE = mgh, where m is mass, g is the acceleration due to gravity (9.8 m/s²), and h is height (50 m).
Step 4: Use the formula for kinetic energy: KE = 0.5mv², where v is the speed we want to find.
Step 5: Set the potential energy equal to the kinetic energy at the lowest point: mgh = 0.5mv².
Step 6: Notice that mass (m) cancels out from both sides of the equation, simplifying it to gh = 0.5v².
Step 7: Rearrange the equation to solve for v: v² = 2gh.
Step 8: Substitute the values for g (9.8 m/s²) and h (50 m) into the equation: v² = 2 * 9.8 * 50.
Step 9: Calculate the right side: v² = 980.
Step 10: Take the square root of both sides to find v: v = sqrt(980).
Step 11: Calculate the square root: v ≈ 31.3 m/s.