Problem Statement
A body is projected horizontally northward with speed $v_0$ at latitude $\lambda$. Find the magnitude and direction of deflection due to the Coriolis force after traveling distance $l$. (Take Earth’s rotation into account.)
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
A body is projected horizontally northward with speed $v_0$ at latitude $\lambda$. Find the magnitude and direction of deflection due to the Coriolis force after traveling distance $l$. (Take Earth’s rotation into account.)
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
A body is projected horizontally northward with speed $v_0$ at latitude $\lambda$. Find the magnitude and direction of deflection due to the Coriolis force after traveling distance $l$. (Take Earth’s rotation into account.)
Given Information
- Initial horizontal speed: $v_0$ (northward), latitude: $\lambda$, travel distance: $l$
- Earth’s angular velocity: $\boldsymbol{\Omega}$ (magnitude $\Omega = 7.27\times10^{-5}$ rad/s)
Physical Concepts & Formulas
Coriolis acceleration: $\mathbf{a}_{\rm Cor} = -2\boldsymbol{\Omega}\times\mathbf{v}$. For northward motion in the Northern Hemisphere, the Coriolis force deflects the body to the right (eastward), with vertical component of $\Omega$ ($= \Omega\sin\lambda$) acting on horizontal velocity.
Step-by-Step Solution
Step 1 — Relevant Coriolis component: For northward velocity $v_0$, the eastward Coriolis acceleration:
$$a_{\rm east} = 2\Omega v_0\sin\lambda$$
Step 2 — Time to travel $l$: $t = l/v_0$
Step 3 — Eastward deflection:
$$\Delta = \frac{1}{2}a_{\rm east}t^2 = \frac{1}{2}\cdot2\Omega v_0\sin\lambda\cdot\frac{l^2}{v_0^2} = \frac{\Omega l^2\sin\lambda}{v_0}$$
Worked Calculation
For $l = 1$ km, $v_0 = 20$ m/s, $\lambda = 60°$: $\Delta = (7.27\times10^{-5})(10^6)(0.866)/20 = 3.1$ m
Answer
$$\boxed{\Delta = \frac{\Omega l^2 \sin\lambda}{v_0}}$$
Physical Interpretation
In the Northern Hemisphere, moving objects deflect to the right (eastward for northward motion). Rivers erode their right banks more in the Northern Hemisphere. Artillery shells fired over long distances require Coriolis correction — a 1 km shot needs several meters of correction at high latitudes.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{\Delta = \frac{\Omega l^2 \sin\lambda}{v_0}}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{\boxed{\Delta = \frac{\Omega l^2 \sin\lambda}{v_0}}}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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