An electron moving along the x axis has a position given by x = 14te^-5t m, where t is in seconds. How far is the electron from the origin when it momentarily stops?
If the electron stops, it means that the velocity has gone to zero.
x = 14te^-5t
dx/dt = 14e^-5t (1 - 5t)
that is zero when t = 1/5
x(.2) = 1.03
To find the distance from the origin when the electron momentarily stops, we need to determine when the velocity of the electron is equal to zero. The velocity is the derivative of the position function with respect to time.
Given the position function x = 14te^(-5t), we can find the velocity by taking the derivative:
v = dx/dt
v = d/dt (14te^(-5t))
To find the derivative, we can use the product rule:
v = 14e^(-5t) + (-5t)(14e^(-5t))
Setting the velocity equal to zero:
0 = 14e^(-5t) + (-5t)(14e^(-5t))
Now, we can solve this equation for t. First, let's divide both sides by 14e^(-5t):
0 = e^(-5t) - 5t(e^(-5t))
Next, we can factor out e^(-5t):
0 = e^(-5t)(1 - 5t)
For this equation to hold true, either e^(-5t) = 0 or (1 - 5t) = 0.
However, e^(-5t) cannot be equal to zero since it is always positive. Thus, we set (1 - 5t) = 0 and solve for t:
1 - 5t = 0
5t = 1
t = 1/5
Therefore, the electron momentarily stops at t = 1/5 seconds.
To find the distance from the origin, we substitute this value of t into the position function:
x = 14t e^(-5t)
x = 14(1/5) e^(-5(1/5))
x = 14/5 e^(-1)
x ≈ 14/5 * 0.368
x ≈ 0.7952 meters
So, the electron is approximately 0.7952 meters from the origin when it momentarily stops.
To find the distance of the electron from the origin when it momentarily stops, we need to determine the point in time when the velocity of the electron becomes zero. At that moment, the electron is at rest, indicating that it has stopped momentarily.
The velocity of an object is the derivative of its position with respect to time. Let's find the velocity function of the electron by taking the derivative of the given position function:
v(t) = d/dt (14te^(-5t))
= 14e^(-5t) - 70te^(-5t)
Now, we need to set the velocity function equal to zero and solve for the time (t) when the electronic momentarily stops:
0 = 14e^(-5t) - 70te^(-5t)
Let's simplify this equation and solve for t:
14e^(-5t) = 70te^(-5t)
Divide both sides by 14e^(-5t) to isolate t:
e^(-5t) = 5t
To solve this equation analytically, we need to use numerical methods or approximation techniques. One approach is to graph both sides of the equation and find their intersection.
Using a graphing calculator or software, plot the functions y = e^(-5t) and y = 5t. Find the intersection point(s), which correspond to the time(s) when the electron momentarily stops.
Once you have the value(s) for t, substitute it back into the position equation x = 14te^(-5t) to find the distance of the electron from the origin.