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Chapter 5: Problem 4

Speedboat A negotiates a curve whose radius is \(120 \mathrm{m}\). Speedboat B negotiates a curve whose radius is \(240 \mathrm{m}\). Each boat experiences the same centripetal acceleration. What is the ratio \(D_{A} / v_{\mathrm{B}}\) of the speeds of the boats?

Short Answer

Expert verified
The ratio of the speeds is \( \frac{\sqrt{2}}{2} \).

Step by step solution

01

Understand Centripetal Acceleration

Centripetal acceleration is given by the formula \( a_c = \frac{v^2}{r} \), where \( v \) is the speed and \( r \) is the radius of the circular path. Since both boats experience the same centripetal acceleration, we have \( \frac{v_A^2}{r_A} = \frac{v_B^2}{r_B} \).
02

Set Up the Equation

Given that \( r_A = 120 \, \mathrm{m} \) and \( r_B = 240 \, \mathrm{m} \), we can equate the centripetal accelerations: \( \frac{v_A^2}{120} = \frac{v_B^2}{240} \).
03

Simplify the Equation

To simplify \( \frac{v_A^2}{120} = \frac{v_B^2}{240} \), we can multiply both sides by 240 to get \( 2v_A^2 = v_B^2 \).
04

Solve for the Ratio of Speeds

From \( 2v_A^2 = v_B^2 \), taking the square root of both sides, we get \( v_B = \sqrt{2}v_A \). Therefore, the ratio \( \frac{v_A}{v_B} = \frac{v_A}{\sqrt{2}v_A} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \).
05

Final Answer

The ratio of the speeds \( \frac{v_A}{v_B} \) is \( \frac{\sqrt{2}}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physics Problems
Physics problems often involve applying mathematical formulas to real-world situations. Understanding the problem is the first important step. For instance, in the given exercise, two speedboats are navigating curves with different radii. It's crucial to decode such problems by identifying what is given and what needs to be resolved. Recognizing key terms like
  • "radius," the distance from the center of the circle to its edge,
  • "centripetal acceleration," the acceleration that keeps an object moving in a circular path,
  • and "ratio," a comparison of two quantities,
These concepts guide the approach to solving the problem.
Physics problems tend to be multi-step processes. By breaking them down into manageable steps like identifying known quantities and relationships, critical thinking and calculation become much simpler. Thus, mastering problem-solving in physics not only aids in answering questions correctly but also builds essential analytical skills.
Speedboat Dynamics
Speedboat dynamics involve understanding how boats move through water, especially when dealing with curves. When a speedboat navigates a curve, it doesn't move in a straight line. Instead, it follows a circular path which requires understanding of circular motion dynamics. The forces involved include:
  • "Thrust," which moves the boat forward,
  • "Drag," which resists its motion through water, and most crucially,
  • "Centripetal Force," which acts towards the center of the circle, keeping the boat in curved motion.
These forces are integrated to maintain a speedboat鈥檚 stability and performance.
The exercise involves understanding centripetal acceleration, which helps determine the speed of the boats as they travel in a curve without losing speed or control. For boats A and B, having the same centripetal acceleration meant analyzing how their speeds relate to the radius of their respective curves, showcasing the importance of radius in speedboat dynamics.
Circular Motion
Circular motion is a fundamental concept in physics and occurs when an object moves in a circle at a constant speed. In the case of the speedboats, they are engaged in uniform circular motion as they navigate curves.Key components in circular motion include:
  • "Radius," the distance from the circle's center to its edge, affects the path's curvature,
  • "Velocity," which has both magnitude (speed) and direction, and
  • "Centripetal Acceleration," critical for keeping an object in a circular path by always pointing towards the circle's center.
The centripetal force is a result of this acceleration.
The formula used for centripetal acceleration, \(a_c = \frac{v^2}{r}\), indicates that acceleration increases with velocity and decreases as the radius increases. In this exercise, understanding how speed and radius interplay helps solve for the ratio of the boats' speeds, reflecting the beauty of mathematical relationships in circular motion. Understanding these relationships reveals how changes in one variable can influence the entire system, leading to comprehensive insights into motion dynamics.

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Most popular questions from this chapter

A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of \(28 \mathrm{m} / \mathrm{s},\) the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius \(=150 \mathrm{m}),\) the block swings toward the outside of the curve. Then the string makes an angle \(\theta\) with the vertical. Find \(\theta\).

A Dangerous Ride. You and your exploration team are stuck on a steep slope in the Andes Mountains in Argentina. A deadly winter storm is approaching and you must get down the mountain before the storm hits. Your path leads you around an extremely slippery, horizontal curve with a diameter of \(90 \mathrm{m}\) and banked at an angle of \(40.0^{\circ}\) relative to the horizontal. You get the idea to unpack the toboggan that you have been using to haul supplies, load your team upon it, and ride it down the mountain to get enough speed to get around the banked curve. You must be extremely careful, however, not to slide down the bank: At the bottom of the curve is a steep cliff. (a) Neglecting friction and air resistance, what must be the speed of your toboggan in order to get around the curve without sliding up or down its bank? Express your answer in \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{m} .\) p.h. (b) You will need to climb up the mountain and ride the toboggan down in order to attain the speed you need to safely navigate the curve (from part (a)), The mountain slope leading into the curve is at an angle of \(30.0^{\circ}\) relative to the horizontal, and the coefficient of kinetic friction between the toboggan and the surface of the slope is \(\left(\mu_{\text {mrantain }}=0.15\right) .\) How far up the mountain (distance along the slope, not elevation) from the curve should you start your ride? Note: the path down the mountain levels off at the bottom so that the toboggan enters the curve moving in the horizontal plane (i.e., in the same plane as the curve).

The hammer throw is a track-and-field event in which a \(7.3-\mathrm{kg}\) ball (the "hammer") is whirled around in a circle several times and released. It then moves upward on the familiar curving path of projectile motion and eventually returns to earth some distance away. The world record for this distance is \(86.75 \mathrm{m},\) achieved in 1986 by Yuriy Sedykh. Ignore air resistance and the fact that the ball is released above the ground rather than at ground level. Furthermore, assume that the ball is whirled on a circle that has a radius of \(1.8 \mathrm{m}\) and that its velocity at the instant of release is directed \(41^{\circ}\) above the horizontal. Find the magnitude of the centripetal force acting on the ball just prior to the moment of release.

A satellite circles the earth in an orbit whose radius is twice the earth's radius. The earth's mass is \(5.98 \times 10^{24} \mathrm{kg}\), and its radius is \(6.38 \times 10^{6} \mathrm{m}\). What is the period of the satellite?

The following table lists data for the speed and radius of three examples of uniform circular motion. Find the magnitude of the centripetal acceleration for each example. $$ \begin{array}{lcc} & \text { Radius } & \text { Speed } \\ \hline \text { Example 1 } & 0.50 \mathrm{m} & 12 \mathrm{m} / \mathrm{s} \\ \hline \text { Example 2 } & \text { Infinitely large } & 35 \mathrm{m} / \mathrm{s} \\ \hline \text { Example 3 } & 1.8 \mathrm{m} & 2.3 \mathrm{m} / \mathrm{s} \\ \hline \end{array} $$
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