A proton with mass m moves in one dimension.The potential-energy function is where and are positive constants. The proton is released from rest a x0 = a/b.
(a) Show that can be written as
U(x) = a
--- [ ( x0/x )^2 - x0/x ]
x0^2
Graph U(x). Calculate U(x0) and thereby locate the point on
the graph. (b) Calculate v(x) the speed of the proton as a function
of position. Graph and give a qualitative description of the
motion. (c) For what value of x is the speed of the proton a maximum? What is the value of that maximum speed? (d) What is
the force on the proton at the point in part (c)? (e) Let the proton be
released instead at x1 = 3a/B Locate the point x1 on the graph of U(x)
Calculate v(x) and give a qualitative description of the
motion. (f ) For each release point ( x = x0 and x = x1 ) what are
the maximum and minimum values of x reached during the motion?
(a) Sure, let's calculate the potential energy function step by step. We have the equation:
U(x) = a * ( (x0/x)^2 - (x0/x) ) / x0^2
Now, let's simplify this equation. First, let's multiply both terms by a/x0^2:
U(x) = (a/x0^2) * ( (x0/x)^2 - (x0/x) )
Next, let's simplify the terms inside the parentheses:
U(x) = (a/x0^2) * ( (x0^2/x^2) - (x0/x) )
U(x) = (a/x0^2) * ( x0^2 - x0*x ) / x^2
Finally, we can simplify this further:
U(x) = a * ( 1 - x/x0 ) / x^2
Now, let's graph this equation.
(b) The speed of the proton, v(x), can be calculated by taking the derivative of the potential energy function with respect to x and then using the equation v(x) = sqrt(2/m * U(x)):
v(x) = sqrt(2/m * (a * (1 - x/x0) / x^2))
We can simplify this further, but let's first graph this equation.
(c) To find the value of x where the speed of the proton is maximum, we need to find the maximum value of v(x). To do this, we can take the derivative of v(x) with respect to x, set it equal to zero, and solve for x.
(d) The force on the proton at the point where the speed is maximum can be calculated using Newton's second law, F = m * a, where a is the acceleration. The acceleration can be found using the equation a(x) = dv(x)/dx.
(e) For x = x1 = 3a/B, we can repeat the same steps as in part (b) to calculate v(x) and give a qualitative description of the motion.
(f) To find the maximum and minimum values of x reached during the motion for each release point, we need to calculate the values of x where the speed of the proton is zero. These correspond to the turning points of the motion.
To solve this problem, let's break it down into steps:
(a) To show that U(x) can be written as mentioned in the question:
We have the potential-energy function U(x) given as:
U(x) = a * [(x0/x)^2 - (x0/x)] / (x0^2)
Let's simplify this expression:
U(x) = a * [(x0^2/x^2) - (x0/x)] / (x0^2)
U(x) = a * [(x0^2 - x0^2/x)] / (x^2 * x0^2)
U(x) = a/x - a/x^3
Graph U(x):
To graph U(x), we need to plot the value of U(x) for different values of x. Start with x = 0 and calculate U(x) for different positive values of x. As x approaches infinity, U(x) approaches zero. Plot these points on a graph and connect them to get a curve.
Calculate U(x0) and locate the point on the graph:
Plug in x = x0 into the expression for U(x):
U(x0) = a/x0 - a/x0^3
This will give you the value of U(x0). Locate this point on the graph.
(b) Calculate v(x) the speed of the proton as a function of position:
The speed of the proton can be calculated by taking the derivative of U(x) with respect to x and then taking the square root:
v(x) = sqrt(-dU(x)/dx)
Calculate this expression to get the speed of the proton as a function of position. Plot this on a graph and describe the motion qualitatively.
(c) To find the value of x at which the speed of the proton is maximum:
Differentiate v(x) with respect to x and find the point where dv(x)/dx = 0. Solve for x to find the position at which the speed is maximum.
To find the maximum speed, substitute the value of x from the previous step into v(x).
(d) The force on the proton at the point in part (c):
The force acting on the proton can be calculated using the relationship:
F(x) = -dU(x)/dx
Differentiate U(x) with respect to x and evaluate it at the point x from part (c) to find the force on the proton.
(e) To calculate v(x) and describe the motion for a release point x1:
Repeat the steps for part (b) but now plug in x = x1 into the expression for U(x) to calculate v(x) for the new release point. Describe the motion qualitatively based on the graph.
(f) For each release point (x = x0 and x = x1), the maximum and minimum values of x reached during the motion:
To find the maximum and minimum values of x reached during the motion, set the expression for U(x) equal to zero and solve for x. This will give you the extreme points on the graph for each release point.