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I'm doing a lab (ballistic pendulum) and I am required to come up with an uncertainty of propagation for the velocity equation given by:

$$V_b = \frac{(m_b + m_c)}{m_b} (2g \Delta h)^{1/2}$$ Where $$\begin{align} V_b &= \text{initial velocity of the ball}\\ m_b &= \text{mass of the ball}\\ m_c &= \text{mass of the catcher}\end{align}$$ Moreover, when I simplify the velocity equation I get in terms of alpha and beta, something like this:

$$\alpha\beta^{-1/2}$$ (assuming that $g$ is constant). How exactly would I relate this to find the uncertainty in the initial velocity $V_b$

vb628
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Joshua Bitton
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1 Answers1

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We propagate uncertainty as follows. If a quantity $y$ is computed as a function of the measured quantities $x_1,x_2,\dots,x_N$ as

$$ y=f(x_1,x_2,\dots,x_N) = f(\mathbf{x})$$

then the squared uncertainty in $y$ at any $\mathbf{x}$ is given by

$$ [\delta y]^2(\mathbf{x}) = \sum_{k=1}^N \bigg(\dfrac{\partial f(\mathbf{x})}{\partial x_k} \: \delta x_k\bigg)^2 $$

where $\delta x_k$ is the uncertainty of each measured quantity $x_k$.

You should do the same using $$ f(m_b,m_c,h) = \frac{(m_b + m_c)}{m_b} (2g h)^{1/2} \text{ .} $$

vb628
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