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Chapter 4: Problem 62

What is the magnitude of the acceleration of a sprinter running at \(10 \mathrm{~m} / \mathrm{s}\) when rounding a turn of radius \(25 \mathrm{~m} ?\)

Short Answer

Expert verified
The magnitude of the acceleration is 4 m/s虏.

Step by step solution

01

Understanding the Problem

We need to find the magnitude of the acceleration of a sprinter who is running around a curve. The given speed of the sprinter is 10 m/s and the curve has a radius of 25 m.
02

Recognize Type of Acceleration

Since the sprinter is rounding a turn, the type of acceleration involved is centripetal acceleration.
03

Centripetal Acceleration Formula

The formula for centripetal acceleration is given by:\[ a_c = \frac{v^2}{r} \]where \(v\) is the velocity, and \(r\) is the radius of the curve.
04

Substitute Values into Formula

Substitute the values into the centripetal acceleration formula:\[ a_c = \frac{(10 \, \text{m/s})^2}{25 \, \text{m}} \]
05

Calculate the Magnitude of Acceleration

Calculate the expression:\[ a_c = \frac{100 \, \text{m}^2/\text{s}^2}{25 \, \text{m}} = 4 \, \text{m/s}^2 \]Therefore, the magnitude of the acceleration is 4 m/s虏.

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Key Concepts

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

Physics Problem-Solving
When approaching physics exercises, especially those involving circular motion and kinematics, there are specific steps you can follow to problem-solve effectively:
  • Start by thoroughly understanding the problem. Identify what is given, such as the speed and radius in our sprinter problem, and what is being asked.
  • Next, recognize the type of motion involved. In scenarios with turns or curves, circular motion principles usually apply.
  • Use appropriate formulas. For circular motion, centripetal acceleration is key, which we've seen through its formula \( a_c = \frac{v^2}{r} \).
  • Substitute the known values into the formula, ensuring your units are consistent.
  • Finally, carry out the calculations to arrive at a solution.
These steps streamline physics problem-solving by allowing you to tackle one aspect of the problem at a time, boosting your confidence.
Circular Motion
Circular motion is a fundamental concept in physics, describing the motion of an object along a circular path. In practical terms, this can be anything from a car taking a curve to a planet orbiting a star. One essential aspect of circular motion is centripetal force, which is what keeps an object moving in a circle rather than in a straight line.
  • This force acts towards the center of the circle, hence its name鈥�'centripetal' meaning 'center-seeking.'
  • Centripetal acceleration, which we calculated for the sprinter, results from this force.
  • The centripetal acceleration formula, \( a_c = \frac{v^2}{r} \), indicates that acceleration increases with higher speeds or tighter curves (smaller radius).
Understanding circular motion is important because it appears frequently in real-life situations and various physics problems.
Kinematics
Kinematics is the branch of physics dedicated to studying motion without considering its causes. It allows us to describe how objects move using terms such as velocity, displacement, and acceleration. In the context of our sprinter problem, kinematics helps us understand how changes in speed and direction affect motion.
  • While the speed of the sprinter remains constant, the change in direction around the curve creates acceleration even without a change in speed magnitude. This is a unique aspect of circular motion.
  • The consistent speed of 10 m/s while turning, combined with the radius of 25 m, allows us to use kinematic equations tailored for circular motion.
  • By knowing the speed and the radius, kinematics allows us to precisely calculate the sprinter's centripetal acceleration, enhancing our understanding of how athletes move efficiently in track events.
Mastering kinematics equips students with practical tools to analyze a wide range of real-world scenarios they might encounter in sports, driving, or even celestial movements.

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Most popular questions from this chapter

A light plane attains an airspeed of \(500 \mathrm{~km} / \mathrm{h}\). The pilot sets out for a destination \(800 \mathrm{~km}\) due north but discovers that the plane must be headed \(20.0^{\circ}\) east of due north to fly there directly. The plane arrives in \(2.00 \mathrm{~h}\). What were the (a) magnitude and (b) direction of the wind velocity?

A rifle that shoots bullets at \(460 \mathrm{~m} / \mathrm{s}\) is to be aimed at a target \(45.7 \mathrm{~m}\) away. If the center of the target is level with the rifle, how high above the target must the rifle barrel be pointed so that the bullet hits dead center?

A dart is thrown horizontally with an initial speed of \(10 \mathrm{~m} / \mathrm{s}\) toward point \(P,\) the bull's-eye on a dart board. It hits at point \(Q\) on the rim, vertically below \(P, 0.19 \mathrm{~s}\) later. (a) What is the distance \(P Q ?\) (b) How far away from the dart board is the dart released?

A cannon located at sea level fires a ball with initial speed \(82 \mathrm{~m} / \mathrm{s}\) and initial angle \(45^{\circ} .\) The ball lands in the water after traveling a horizontal distance \(686 \mathrm{~m} .\) How much greater would the horizontal distance have been had the cannon been \(30 \mathrm{~m}\) higher?

A woman rides a carnival Ferris wheel at radius \(15 \mathrm{~m}\), completing five turns about its horizontal axis every minute. What are (a) the period of the motion, the (b) magnitude and (c) direction of her centripetal acceleration at the highest point, and the (d) magnitude and (e) direction of her centripetal acceleration at the lowest point?
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