The equation to find the distance of the end of the spring below equilibrium in terms of time can be written as D(t) = D0 * e^(-kt), where D0 is the initial amplitude, k is the rate of amplitude decrease, and t is the time elapsed. Using the given information, we can solve for k and substitute it into the equation to find the final equation.
The equation that describes the distance, D(t), of the end of the spring below equilibrium in terms of time, t, can be found using the formula for exponential decay:
D(t) = D0 * e-kt
Where:
D0 is the initial amplitude of the spring
k is the rate at which the amplitude decreases
t is the time
In this case, the initial amplitude is 9 cm and after 3 seconds the amplitude has decreased to 8 cm. Therefore, we can substitute these values into the equation to find the value of k:
D(0) = 9 cm
D(3) = 8 cm
Using these values in the equation, we can solve for k:
8 cm = 9 cm * e-3k
Dividing both sides by 9 cm:
e-3k = 8/9
Taking the natural logarithm of both sides:
-3k = ln(8/9)
Dividing both sides by -3:
k = -ln(8/9) / 3
Now we can use this value of k to find the equation for D(t):
D(t) = 9 cm * e(-ln(8/9) / 3)t