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Chapter 2: Problem 28

(a) What is the magnitude of the average acceleration of a skier who, starting from rest, reaches a speed of \(8.0 \mathrm{m} / \mathrm{s}\) when going down a slope for \(5.0 \mathrm{s} ?\) (b) How far does the skier travel in this time?

Short Answer

Expert verified
(a) The average acceleration is 1.6 m/s虏. (b) The skier travels 20 meters.

Step by step solution

01

Analyze the Given Information

The problem provides the initial speed of the skier as 0 m/s (at rest), the final speed as 8.0 m/s, and the time taken to reach this speed as 5.0 seconds. We need to find the average acceleration and the distance traveled.
02

Use the Formula for Average Acceleration

Average acceleration is defined as the change in velocity divided by the time taken to make that change. The formula is:\[a = \frac{v_f - v_i}{t}\]where \(v_f = 8.0\, \text{m/s}\), \(v_i = 0\, \text{m/s}\), and \(t = 5.0\, \text{s}\).
03

Calculate the Average Acceleration

Substitute the given values into the formula:\[a = \frac{8.0\, \text{m/s} - 0\, \text{m/s}}{5.0\, \text{s}} = \frac{8.0\, \text{m/s}}{5.0\, \text{s}} = 1.6\, \text{m/s}^2\]The magnitude of the average acceleration is \(1.6\, \text{m/s}^2\).
04

Use the Kinematic Equation for Distance

To find the distance traveled, we use the kinematic equation:\[ d = v_i t + \frac{1}{2} a t^2\]Given \(v_i = 0\, \text{m/s}\), \(a = 1.6\, \text{m/s}^2\), and \(t = 5.0\, \text{s}\).
05

Calculate the Distance Traveled

Substitute the known values into the formula:\[d = 0\cdot5 + \frac{1}{2} \cdot 1.6 \cdot (5)^2 = \frac{1}{2} \cdot 1.6 \cdot 25 = 20 \, \text{m}\]The skier travels 20 meters in 5 seconds.

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Key Concepts

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

Kinematic Equations
Kinematic equations are the backbone of understanding motion in physics. These equations relate the five key components of motion: displacement, initial velocity, final velocity, acceleration, and time.
They are crucial for predicting the behavior of moving objects and solving various physics problems like the one with the skier. There are several key kinematic equations, including:
  • Final velocity equation: \( v_f = v_i + at \)
  • Displacement with constant acceleration equation: \( d = v_i t + \frac{1}{2} a t^2 \)
  • Another form of displacement equation: \( d = \frac{(v_i + v_f)}{2} t \)
  • Velocity squared equation: \( v_f^2 = v_i^2 + 2ad \)
In the skier's problem, we applied these equations to find both acceleration and distance. The key takeaway is understanding how velocity changes over time and using the correct formula based on the known information.
Velocity
Velocity is more than just speed; it's a vector quantity, which means it has both magnitude and direction. In physics, knowing the direction of movement is as important as knowing how fast an object is moving.Velocity tells us how quickly an object changes its position. The difference between velocity and speed is crucial:
  • Speed: It's simply how fast something is moving, regardless of its direction.
  • Velocity: Includes both the speed and direction of the motion, thus can be positive, negative, or zero.
In the context of our skier problem, the skier starts from rest, which means their initial velocity is \(0 \text{ m/s}\). Eventually, the skier reaches a velocity of \(8.0 \text{ m/s}\) downward on the slope. This change in velocity is what allowed us to calculate the average acceleration, showcasing the importance of velocity in analyzing motion.
Time
Time is a critical factor in analyzing motion, often considered alongside other variables like velocity and acceleration. It allows us to understand how quickly events occur.The role of time becomes especially clear when solving physics problems like kinematic equations where:
  • It acts as a measure of how long a change takes place. For instance, in our skier problem, it took 5 seconds for the skier to reach a speed of \(8.0 \text{ m/s}\).
  • The changes investigated over a time interval include changes in velocity and position, allowing us to calculate things like average acceleration and displacement.
By understanding the time component in these equations, one can gauge not just the magnitude of changes in motion but also their tempo, making it essential for predicting and managing real-world scenarios.

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Most popular questions from this chapter

From her bedroom window a girl drops a water-filled balloon to the ground, \(6.0 \mathrm{m}\) below. If the balloon is released from rest, how long is it in the air?

A motorcycle has a constant acceleration of \(2.5 \mathrm{m} / \mathrm{s}^{2} .\) Both the velocity and acceleration of the motorcycle point in the same direction. How much time is required for the motorcycle to change its speed from (a) 21 to \(31 \mathrm{m} / \mathrm{s},\) and (b) 51 to \(61 \mathrm{m} / \mathrm{s} ?\)

In a historical movie, two knights on horseback start from rest \(88.0 \mathrm{m}\) apart and ride directly toward each other to do battle. Sir George's acceleration has a magnitude of \(0.300 \mathrm{m} / \mathrm{s}^{2},\) while Sir Alfred's has a magnitude of \(0.200 \mathrm{m} / \mathrm{s}^{2} .\) Relative to Sir George's starting point, where do the knights collide?

A jogger accelerates from rest to \(3.0 \mathrm{m} / \mathrm{s}\) in \(2.0 \mathrm{s}\). A car accelerates from 38.0 to \(41.0 \mathrm{m} / \mathrm{s}\) also in \(2.0 \mathrm{s}\). (a) Find the acceleration (magnitude only) of the jogger. (b) Determine the acceleration (magnitude only) of the car. (c) Does the car travel farther than the jogger during the 2.0 s? If so, how much farther?

You step onto a hot beach with your bare feet. A nerve impulse, generated in your foot, travels through your nervous system at an average speed of \(110 \mathrm{m} / \mathrm{s} .\) How much time does it take for the impulse, which travels a distance of \(1.8 \mathrm{m},\) to reach your brain?
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