Error Analysis- Deriving Equations

[bchan907]

Homework Statement

In this experiment, the acceleration due to gravity is related to the time, t, it takes the ball to fall to a distance , y, by Eq. (3). With this equation and Eq. (2), show that the fractional error, ∆g/g is ( (∆y/y)^2 + (2∆t/t)^2 ) ^1/2

Homework Equations

Eq. (3) = g = 2y/t^2

Eq. (2) = ∆W/W = ( (n∆a/a)^2 + (m∆b/b)^2 + (p∆c/c)^2 ) ^1/2

The Attempt at a Solution

I am totally confused on how to do this problem. Could anyone help? Thanks!

 

[anathema]

I would like to clarify the equations given in the forum post. Eq. (3) is the equation for the acceleration due to gravity, which can be derived from Newton's second law (F=ma) and the equation for the distance traveled by a falling object (y=1/2gt^2). Eq. (2) is a general equation for calculating the fractional error in a quantity, where n, m, and p represent the number of times that the quantity is multiplied, divided, or raised to a power, respectively, and ∆a, ∆b, and ∆c represent the uncertainties in those quantities.

To solve this problem, we can start by substituting Eq. (3) into Eq. (2) for g, giving us:

∆g/g = ( (∆y/y)^2 + (2∆t/t)^2 )^1/2

This shows that the fractional error in g is equal to the square root of the sum of the squares of the fractional errors in y and t. This makes sense because both y and t are directly related to g in Eq. (3), so any uncertainties in those quantities will affect the calculated value of g.

To better understand this relationship, let's consider a specific example. Let's say we are measuring the acceleration due to gravity using a ball that is dropped from a height of 1 meter. We measure the time it takes for the ball to fall to be 0.5 seconds with an uncertainty of ±0.1 seconds. We also measure the distance traveled by the ball to be 0.25 meters with an uncertainty of ±0.05 meters. Using these values, we can calculate the fractional error in g as:

∆g/g = ( (0.05/0.25)^2 + (2(0.1/0.5))^2 )^1/2 = (0.04 + 0.08)^1/2 = 0.1

This means that the uncertainty in our measurement of g is 10% of the calculated value. This shows the importance of minimizing uncertainties in our measurements, as they can greatly affect the accuracy of our results.

In conclusion, the equation given in the forum post is correct and can be derived from the fundamental equations for acceleration due to gravity and calculating fractional errors. It is important for scientists to understand and apply these equations in order to accurately report their findings