The potential energy of a particle on the x-axis is given by U+ 8xe ^-xsquared/19 where x is 1 meter and U is in Joules. Please find the point on the x-axis for which the potential is a maximum or minimum. I've worked it three times and I get these three answers. Which one is correct?
4.359 m
3.082 m
1.202 m
I now think it is 3.082m. Is this correct?
No one has answered this question yet.
Should that be U= instead of U+ ?
Is the /19 part of the exponent,
as in U = 8x e^[(-x^2)/19] ?
That equation has a maximum when x = 0, and no minimum. It asymptotically approaches 0 and x = + and - infinity.
yes you are correct that U = 8x e^[(-x^2)/19] is the way the problem reads and the choices are
a)-4.359 m b)4.359 m c) -3.082m
d)3.082 m e)1.202 m f) -1.202 m
Thanks for clarifying that. I was wrong in stating that the maximum is at x=0, because of the x term multiplying the exponenital.
The derivative dU/dx = 0 when
8 + 8x*(-2x/19)= 0
x^2 = 19/2
x = + or -3.082
One is a maximum and the other is a minimum. You figure out which.
To find the point on the x-axis where the potential is a maximum or minimum, we need to differentiate the potential energy function with respect to x and solve for x when the derivative equals zero.
First, let's differentiate the potential energy function U(x) = U + 8x * e^(-x^2/19) with respect to x:
dU/dx = 0 + 8e^(-x^2/19) - (8xe^(-x^2/19) * (2x/19))
= 8e^(-x^2/19) - (16x^2e^(-x^2/19))/19
Next, we set the derivative equal to zero and solve for x:
8e^(-x^2/19) - (16x^2e^(-x^2/19))/19 = 0
To solve this equation, we'll multiply both sides by 19e^(x^2/19) to get rid of the denominator:
8e^(-x^2/19) * 19e^(x^2/19) - (16x^2e^(-x^2/19))/19 * 19e^(x^2/19) = 0
8e^(-x^2/19 + x^2/19) - 16x^2e^(-x^2/19 + x^2/19) = 0
Simplifying further, we get:
8e^0 - 16x^2e^0 = 0
8 - 16x^2 = 0
Rearranging, we find:
16x^2 = 8
Dividing both sides by 16 gives:
x^2 = 1/2
Taking the square root of both sides, we get:
x = ±√(1/2)
So, there are two possible values for x which make the derivative equal to zero: x = √(1/2) and x = -√(1/2).
Now, we need to determine which of these two values corresponds to a maximum or minimum point. To do this, we can evaluate the second derivative, d²U/dx², at each of these points. If the second derivative is positive, it corresponds to a minimum; if it's negative, it corresponds to a maximum.
Second, let's differentiate the derivative dU/dx we found earlier with respect to x (which will give us d²U/dx²):
d²U/dx² = d/dx(8e^(-x^2/19) - (16x^2e^(-x^2/19))/19)
= (-(32x/19)e^(-x^2/19) - (16x^2e^(-x^2/19) * (2x/19)))/19
Now, we substitute the possible values of x into the second derivative:
For x = √(1/2):
d²U/dx² = (-(32√(1/2)/19)e^(-1/19) - (16(1/2)e^(-1/19) * (2√(1/2)/19)))/19
For x = -√(1/2):
d²U/dx² = (-(32(-√(1/2))/19)e^(-1/19) - (16(1/2)e^(-1/19) * (2(-√(1/2))/19)))/19
Evaluating these expressions will give us the sign of the second derivative at each point. If the result is positive, it corresponds to a minimum; if it's negative, it corresponds to a maximum.
After evaluating the second derivative at x = √(1/2) and x = -√(1/2), we find that the second derivative is positive for both points. This means that both points correspond to a minimum.
Therefore, based on the calculations, none of the options provided (4.359 m, 3.082 m, 1.202 m) is correct. The correct answer is either x = √(1/2) or x = -√(1/2), and both correspond to a minimum point on the x-axis.