Question

you are measuring the average speed of your well - trained dog. the dog runs to the end of a field and back to you. you measure the length of the field, l, and the round - trip time, t, with their uncertainties δl and δt, to calculate the average speed of the dog: v = 2l/t. using the multiplication and division rule for propagation of uncertainties, select the expression for the absolute uncertainty in the average speed.
form: r = xy or r = x/y
rule: δr/r = δx/x+ δy/y
a) δv=(δl/l+ δt/t)
b) δv=(2δl/l+ δt/t)v
c) δv=(δl/l+ δt/t)v
d) δv=(δl/l - δt/t)v
e) δv=(2δl/l - δt/t)v

Explanation:

Step1: Identify variables in the speed formula

The speed formula is $v = \frac{2L}{t}$, where $x = 2L$ and $y=t$.

Step2: Apply uncertainty - propagation rule

The rule for $r=\frac{x}{y}$ is $\frac{\Delta r}{r}=\frac{\Delta x}{x}+\frac{\Delta y}{y}$. Here, $\Delta x = 2\Delta L$ (since $x = 2L$) and $\Delta y=\Delta t$. So, $\frac{\Delta v}{v}=\frac{2\Delta L}{L}+\frac{\Delta t}{t}$.

Step3: Solve for $\Delta v$

Cross - multiply the above equation by $v$ to get $\Delta v=(\frac{2\Delta L}{L}+\frac{\Delta t}{t})v$.

Answer:

B. $\Delta v=(\frac{2\Delta L}{L}+\frac{\Delta t}{t})v$