Suppose that you measure the velocity of an object by measuring that it takes 1 second to travel 1.2 meters. The measurement error is .001 meters in distance, and the error in time is .01 second. What is the absolute value of the error in the linear approximation for the velocity?
|∆v| ≤ ???
when quantities are multiplied or divided, the PERCENT errors are added
To find the absolute value of the error in the linear approximation for the velocity, we need to consider the errors in distance and time measurements.
The linear approximation for velocity is given by the formula:
v = ∆d / ∆t
where v is the velocity, ∆d is the change in distance, and ∆t is the change in time.
In this case, we have ∆d = 1.2 meters and ∆t = 1 second. Let's denote the errors in distance and time as ∆d_error and ∆t_error, respectively.
To find the error in the linear approximation, we can use the formula:
∆v = |∆d / ∆t| * ∆t_error + |∆d / ∆t|^2 * ∆d_error
Substituting the given values into the formula, we get:
∆v = |1.2 / 1| * 0.01 + |1.2 / 1|^2 * 0.001
Simplifying, we have:
∆v = |1.2| * 0.01 + |1.2|^2 * 0.001
∆v = 1.2 * 0.01 + 1.44 * 0.001
∆v = 0.012 + 0.00144
∆v = 0.01344
Therefore, the absolute value of the error in the linear approximation for the velocity is 0.01344.