The momentum $p$ of an object is given by the formula:

$p = m \cdot v$

where $m$ is the mass and $v$ is the velocity.

Step 1: Convert the velocity to SI units (m/s)

Given:

• $v = (132 \pm 3) \, \text{cm/s}$

Convert to meters per second: $v = 132 \, \text{cm/s} = 1.32 \, \text{m/s}$ and the uncertainty: $\Delta v = 3 \, \text{cm/s} = 0.03 \, \text{m/s}$

Step 2: Apply the given mass

The mass is already in SI units: $m = (1.57 \pm 0.08) \times 10^{-12} \, \text{kg}$

Step 3: Calculate the momentum

Now calculate the momentum using the central values of $m$ and $v$: $p = m \cdot v = (1.57 \times 10^{-12}) \cdot (1.32) = 2.0724 \times 10^{-12} \, \text{kg m/s}$

Step 4: Calculate the uncertainty

To find the uncertainty in the momentum, use the formula for the propagation of uncertainties in multiplication:

$\frac{\Delta p}{p} = \sqrt{\left(\frac{\Delta m}{m}\right)^2 + \left(\frac{\Delta v}{v}\right)^2}$

Substitute the uncertainties: $\frac{\Delta p}{p} = \sqrt{\left(\frac{0.08}{1.57}\right)^2 + \left(\frac{0.03}{1.32}\right)^2}$ $\frac{\Delta p}{p} = \sqrt{(0.05096)^2 + (0.02273)^2} = \sqrt{0.002597 + 0.000517} = \sqrt{0.003114} \approx 0.0558$

So the relative uncertainty is approximately $0.0558$, or $5.58\%$.

Step 5: Final uncertainty in momentum

Now calculate the absolute uncertainty in $p$: $\Delta p = 0.0558 \times 2.0724 \times 10^{-12} = 0.1156 \times 10^{-12} = 1.16 \times 10^{-13} \, \text{kg m/s}$

Final Answer

The magnitude of the momentum is: $p = (2.07 \pm 0.12) \times 10^{-12} \, \text{kg m/s}$

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Would you like more details or have any questions?

Here are some related questions:

1. How does momentum change if velocity doubles while mass remains constant?
2. What is the significance of momentum conservation in collisions?
3. How does uncertainty propagation apply to other physical quantities?
4. How is momentum used in understanding Newton’s Second Law?
5. What role does momentum play in rocket propulsion?

Tip: When working with uncertainties, always combine them using the appropriate formulas for the mathematical operations involved (e.g., addition, multiplication).