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Chapter 7: Problem 66

You have a summer job at your university's zoology department, where you'll be working with an animal behavior expert. She's assigned you to study videos of different animals leaping into the air. Your task is to compare their power outputs as they jump. You'll have the mass \(m\) of each animal from data collected in the field. From the videos, you'll be able to measure both the vertical distance \(d\) over which the animal accelerates when it pushes off the ground and the maximum height \(h\) it reaches. Your task is to find an algebraic expression for power in terms of these parameters.

Short Answer

Expert verified
The power output of the animals leaping into the air can be represented using the formula \(P=mgd/sqrt(2d/g)\), where \(m\) is the mass of the animal, \(g\) is the gravitational acceleration, \(d\) is the distance over which the animal accelerates, and \(h\) is the maximum height reached by the animal.

Step by step solution

01

Define the parameters

First, we'll define the parameters. The mass of the animal is \(m\), the distance over which it accelerates is \(d\), and the maximum height reached is \(h\). The goal is to find power, or work done over time.
02

Derive Force and Work

Next, derive the force exerted by the animal during the jump. It accelerates over a distance \(d\) when it jumps, so the force is equal to mass times acceleration (\(F=ma\)). Also, work (\(W\)) is defined as force times distance (\(W=Fd\)). So, \(W=mad\).
03

Derive Acceleration

In order to calculate the acceleration, we will use the equation \(v^2 = 2gh\), where \(v\) is velocity at the maximum height, which is 0, \(g\) is gravitational acceleration, and \(h\) is the maximum height. Therefore, \(a = g\), which is equivalent to \(9.8 m/s^2\).
04

Substitute Force, Work and Acceleration in Power Formula

Now we substitute \(F=ma\), \(W=Fd=mad\), and \(a=g\) into the power \(P\) equation, which is \(P=W/t\). Power is given by \(P=mgd/t\).
05

Derive time

We use the equations of motion to derive time. The body is accelerating up to a height \(d\), so using \(h=ut + 1/2gt^2\) where \(u=0\) (initial velocity of the animal), we find that \(t = sqrt(2d/g)\).
06

Finalize the power formula

Substitute the derived time expression \(t = sqrt(2d/g)\) back into the power equation. This gives the formula for power as \(P=mgd/sqrt(2d/g)\). This is the final expression for power in terms of \(m\), \(g\), \(d\), and \(h\).

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