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Chapter 4: Problem 96

A person in a kayak starts paddling, and it accelerates from 0 to \(0.60 \mathrm{~m} / \mathrm{s}\) in a distance of \(0.41 \mathrm{~m} .\) If the combined mass of the person and the kayak is \(73 \mathrm{~kg},\) what is the magnitude of the net force acting on the kayak?

Short Answer

Expert verified
The net force is approximately 32.05 N.

Step by step solution

01

Identify the Known Variables

We are given the initial velocity \( v_i = 0 \; \mathrm{m/s} \), final velocity \( v_f = 0.60 \; \mathrm{m/s} \), distance \( s = 0.41 \; \mathrm{m} \), and the total mass \( m = 73 \; \mathrm{kg} \). We need to find the net force \( F \).
02

Use the Kinematic Equation to Find Acceleration

The kinematic equation that relates velocity, acceleration, and distance is:\[v_f^2 = v_i^2 + 2as\]Solving for acceleration \( a \):\[a = \frac{v_f^2 - v_i^2}{2s}\]Substitute the given values:\[a = \frac{(0.60)^2 - 0^2}{2 \cdot 0.41} = \frac{0.36}{0.82} \approx 0.439 \; \mathrm{m/s^2}\]
03

Calculate the Net Force Using Newton's Second Law

Newton's Second Law states that \( F = ma \). Now, substitute the known values:\[F = 73 \cdot 0.439 = 32.047 \; \mathrm{N}\]
04

Provide the Magnitude of the Net Force

The magnitude of the net force acting on the kayak is approximately \( 32.05 \; \mathrm{N} \) when rounded to two decimal places.

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Key Concepts

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

Kinematic Equations and Motion
Kinematic equations are a set of mathematical formulas used to predict the motion of an object moving along a straight path. In the context of the exercise, we're interested in how these equations help us find acceleration during the kayak's motion. The specific equation used here is:\[v_f^2 = v_i^2 + 2as\]
  • In this formula, \(v_f\) is the final velocity, \(v_i\) is the initial velocity, \(a\) is acceleration, and \(s\) is the distance covered.
  • We're given that the kayak starts from rest, which simplifies the equation since \(v_i = 0\).
  • The equation effectively ties together the starting conditions of an object, its final speed, the acceleration acting on it, and the distance traveled.
By using this equation, you can understand how objects speed up or slow down over a set distance, which is crucial for not just kayaks, but for any object in motion.
Understanding Acceleration Calculation
Acceleration is the rate at which an object's velocity changes with time. To calculate it, we must have certain initial conditions:
  • The initial and final velocities
  • The distance over which the change occurs
Here, the kinematic equation was manipulated to solve for acceleration:\[a = \frac{v_f^2 - v_i^2}{2s}\]
  • By plugging in the numbers from our exercise, \(v_i = 0\), and distance \(s = 0.41\, \text{m}\), we found the acceleration \(a\) to be approximately \(0.439 \, \mathrm{m/s^2}\).
  • Acceleration is a vector quantity, meaning it has both magnitude and direction.
This calculation provides insight into how quickly the kayak is able to change speed over the given distance.
Net Force Determination Using Newton's Second Law
Newton's Second Law of Motion is a powerful principle that connects force, mass, and acceleration. It is summarized in the formula:\[F = ma\]
  • Where \(F\) is the net force applied to an object, \(m\) is the mass, and \(a\) is the acceleration.
  • In our kayak problem, the mass \(m\) is 73 kg, and we've calculated the acceleration to be \(0.439 \, \mathrm{m/s^2}\).
Using these values, we determined the net force acting on the kayak by substituting them into Newton's equation:\[F = 73 \times 0.439 \approx 32.05 \, \text{N}\]
  • This result demonstrates how the applied force enables the kayak to accelerate along the water.
  • By understanding this relationship, you can gain a deeper appreciation of how forces affect motion in real-world scenarios.

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Most popular questions from this chapter

Problem 77 The drawing shows Robin Hood (mass \(=77.0 \mathrm{~kg}\) ) about to escape from a dangerous situation. With one hand, he is gripping the rope that holds up a chandelier (mass \(=195\) kg). When he cuts the rope where it is tied to the floor, the chandelier will fall, and he will be pulled up toward a balcony above. Ignore the friction between the rope and the beams over which it slides, and find (a) the acceleration with which Robin is pulled upward and (b) the tension in the rope while Robin escapes.

A 15 -g bullet is fired from a rifle. It takes \(2.50 \times 10^{-3} \mathrm{~s} \mathrm{~s}\) for the bullet to travel the length of the barrel, and it exits the barrel with a speed of \(715 \mathrm{~m} / \mathrm{s}\). Assuming that the acceleration of the bullet is constant, find the average net force exerted on the bullet.

A \(95.0-\mathrm{kg}\) person stands on a scale in an elevator. What is the apparent weight when the elevator is (a) accelerating upward with an acceleration of \(1.80 \mathrm{~m} / \mathrm{s}^{2},\) (b) moving upward at a constant speed, and (c) accelerating downward with an acceleration of \(1.30 \mathrm{~m} / \mathrm{s}^{2}\) ?

A car is towing a boat on a trailer. The driver starts from rest and accelerates to a velocity of \(+11 \mathrm{~m} / \mathrm{s}\) in a time of \(28 \mathrm{~s}\). The combined mass of the boat and trailer is 410 kg. The frictional force acting on the trailer can be ignored. What is the tension in the hitch that connects the trailer to the car?

In the amusement park ride known as Magic Mountain Superman, powerful magnets accelerate a car and its riders from rest to \(45 \mathrm{~m} / \mathrm{s}\) (about \(100 \mathrm{mi} / \mathrm{h}\) ) in a time of \(7.0 \mathrm{~s}\). The mass of the car and riders is \(5.5 \times 10^{3} \mathrm{~kg} .\) Find the average net force exerted on the car and riders by the magnets.
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