Subfield: Mechanics

- Proton mass, $m_{p}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Neutron mass, $m_{n}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Electron mass, $m_{e}=9.11 \cdot 10^{-31} \mathrm{~kg}$
- Avogadro's constant, $N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1}$
- Universal gas constant, $R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$
- Boltzmann's constant, $k_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K}$
- Electron charge magnitude, $e=1.60 \cdot 10^{-19} \mathrm{C}$
- 1 electron volt, $1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J}$
- Speed of light, $c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}$
- Universal Gravitational constant,

$$
G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2}
$$

- Solar Mass

$$
M_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg}
$$

- Acceleration due to gravity, $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$
- 1 unified atomic mass unit,

$$
1 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2}
$$

- Planck's constant,

$$
h=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s}
$$

- Permittivity of free space,

$$
\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right)
$$

- Coulomb's law constant,

$$
k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2}
$$

- Permeability of free space,

$$
\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}
$$

- Magnetic constant,

$$
\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A}
$$

- 1 atmospheric pressure,

$$
1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa}
$$

- Wien's displacement constant, $b=2.9$. $10^{-3} \mathrm{~m} \cdot \mathrm{K}$
- Stefan-Boltzmann constant,

$$
\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}
$$

Question: Jerry spots a truckload of his favourite golden yellow Swiss cheese being transported on a cart moving at a constant velocity $v_{0}=5 \mathrm{~m} / \mathrm{s} \hat{i}$ along the x-axis, which is initially placed at $(0,0)$. Jerry, driven by desire immediately starts pursuing the cheese-truck in such a way that his velocity vector always points towards the cheese-truck; however, Jerry is smart and knows that he must maintain a constant distance $\ell=10 \mathrm{~m}$ from the truck to avoid being caught by anyone, no matter what. Note that Jerry starts at coordinates $(0, \ell)$.


Let the magnitude of velocity (in $\mathrm{m} / \mathrm{s}$ ) and acceleration (in $\mathrm{m} / \mathrm{s}^{2}$ ) of Jerry at the moment when the (acute) angle between the two velocity vectors is $\theta=60^{\circ}$ be $\alpha$ and $\beta$ respectively. Compute $\alpha^{2}+\beta^{2}$.

Solution:
If the distance between Jerry and the cheese truck is constant, then Jerry moves in circle of radius $\ell$ in the reference frame of the cheese truck. There is no radial component of Jerry's velocity in this reference frame, so we must have $\alpha=v_{0} \cos \theta=\frac{5}{2}$. In this case, the tangential velocity is $v_{0} \sin \theta$. Furthermore, the radial acceleration in this frame is given by the centripetal acceleration which is $\frac{\left(v_{0} \sin \theta\right)^{2}}{\ell}=\frac{v_{0}^{2} \sin ^{2} \theta}{\ell}$. The tangential acceleration is

$$
\frac{d}{d t}\left(v_{0} \sin \theta\right)=v_{0} \cos \theta \cdot \frac{d \theta}{d t}=v_{0} \cos \theta \cdot \frac{-v_{0} \sin \theta}{\ell}=-\frac{v_{0}^{2} \sin \theta \cos \theta}{\ell}
$$

The vector sum of these accelerations has magnitude

$$
\beta=\sqrt{\left(\frac{v_{0}^{2} \sin ^{2} \theta}{\ell}\right)^{2}+\left(\frac{v_{0}^{2} \sin \theta \cos \theta}{\ell}\right)^{2}}=\frac{v_{0}^{2} \sin \theta}{\ell}=\frac{5 \sqrt{3}}{4} .
$$

The final answer is $\alpha^{2}+\beta^{2}=10.9375$.

Final answer: 10.9375