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Chapter 7: Problem 48

A particle with total energy \(3.5 \mathrm{J}\) is trapped in a potential well described by \(U=7.0-8.0 x+1.7 x^{2},\) where \(U\) is in joules and \(x\) in meters. Find its turning points.

Short Answer

Expert verified
The turning points can be found by solving the equation \( 1.7x^2 - 8.0x + 3.5 = 0 \) using the quadratic formula.

Step by step solution

01

Write Down the Equations

First, write down the equation for the potential energy U of the particle in the well, given by \( U = 7.0 - 8.0x + 1.7x^2 \). Also write down the equation equating the total energy E of the particle with its potential energy: \( E = U \). Hence, \( 3.5 = 7.0 - 8.0x + 1.7x^2 \).
02

Rearrange The Equation

Next, we need to make this a standard quadratic equation. Start by subtracting 3.5 from both sides to get it equal to zero. The equation now looks like this: \( 1.7x^2 - 8.0x + 3.5 = 0 \).
03

Solve for x (Finding the Turning Points)

To find x, the roots of the quadratic equation, you can use the quadratic formula \( x = [-b 卤 sqrt(b^2 - 4ac)] / (2a) \), where a=1.7, b=-8.0 and c=3.5. Substituting these values in will give the values of x, which are the turning points of the particle in the well.

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