Questions: Speedboat A negotiates a curve whose radius is 112 m. Speedboat B negotiates a curve whose radius is 240 m. Each boat experiences the same centripetal acceleration. What is the ratio v / vB of the speeds of the boats? The weight of an object is the same on two different planets. The mass of planet A is only eighty percent that of planet B. Find the ratio rA / rB of the radii of the planets.

Speedboat A negotiates a curve whose radius is 112 m. Speedboat B negotiates a curve whose radius is 240 m. Each boat experiences the same centripetal acceleration. What is the ratio v / vB of the speeds of the boats?

The weight of an object is the same on two different planets. The mass of planet A is only eighty percent that of planet B. Find the ratio rA / rB of the radii of the planets.

Transcript text: Speedboat A negotiates a curve whose radius is 112 m . Speedboat B negotiates a curve whose radius is 240 m . Each boat experiences the same centripetal acceleration. What is the ratio $\mathrm{v} / \mathrm{l} \mathrm{v}_{\mathrm{B}}$ of the speeds of the boats? The weight of an object is the same on two different planets. The mass of planet A is only eighty percent that of planet B . Find the ratio $r_{A} / r_{B}$ of the radii of the planets.

Solution

Solution Steps

Step 1: Understand the Relationship for Centripetal Acceleration

The centripetal acceleration $ a_c $ for an object moving in a circle of radius $ r $ with speed $ v $ is given by the formula $ a_c = \frac{v^2}{r} $.

Step 2: Set Up the Equation for Equal Centripetal Accelerations

Since both speedboats experience the same centripetal acceleration, we have: \[ \frac{v_A^2}{r_A} = \frac{v_B^2}{r_B} \] where $ r_A = 112 \, \text{m} $ and $ r_B = 240 \, \text{m} $.

Step 3: Solve for the Ratio of Speeds

Rearrange the equation to find the ratio of the speeds: \[ \frac{v_A^2}{v_B^2} = \frac{r_A}{r_B} \]
Take the square root of both sides to find: \[ \frac{v_A}{v_B} = \sqrt{\frac{r_A}{r_B}} \]
Substitute the given radii: \[ \frac{v_A}{v_B} = \sqrt{\frac{112}{240}} \]

Step 4: Simplify the Expression

Simplify the fraction under the square root: \[ \frac{v_A}{v_B} = \sqrt{\frac{7}{15}} \]

Step 5: Understand the Relationship for Gravitational Force

The gravitational force $ F $ on an object is given by $ F = \frac{G \cdot m \cdot M}{r^2} $, where $ M $ is the mass of the planet, $ m $ is the mass of the object, $ r $ is the radius of the planet, and $ G $ is the gravitational constant.

Step 6: Set Up the Equation for Equal Weights

Since the weight of the object is the same on both planets, we have: \[ \frac{G \cdot m \cdot M_A}{r_A^2} = \frac{G \cdot m \cdot M_B}{r_B^2} \]

Step 7: Solve for the Ratio of Radii

Simplify the equation: \[ \frac{M_A}{r_A^2} = \frac{M_B}{r_B^2} \]
Given $ M_A = 0.8 \cdot M_B $, substitute and solve for the ratio of radii: \[ \frac{0.8 \cdot M_B}{r_A^2} = \frac{M_B}{r_B^2} \]
Cancel $ M_B $ and rearrange: \[ \frac{r_A^2}{r_B^2} = 0.8 \]
Take the square root of both sides: \[ \frac{r_A}{r_B} = \sqrt{0.8} \]