Classical mechanics

A football is kicked from the ground at $20\ \mathrm{m/s}$, at $45^\circ$ above the horizontal. Ignoring air resistance and taking $g = 10\ \mathrm{m/s^2}$, how far away (in metres) does it land?

An astronaut has mass $70$ kg. On the Moon ($g \approx 1.6\ \mathrm{m/s^2}$), which statement is correct?

A hockey puck slides in a straight line at constant speed across perfectly frictionless ice. What is the net horizontal force on it?

A mass of $0.5$ kg hangs on a spring of stiffness $k = 18\ \mathrm{N/m}$. What is the angular frequency $\omega$ of its small oscillations, in rad/s?

An asteroid orbits the Sun on an ellipse with semi-major axis $4$ AU (Earth's is $1$ AU, with period $1$ year). Using Kepler's third law, what is its period in years?

A particle moves under a central force $\mathbf F = f(\mathbf r, t)\,\hat{\mathbf r}$ (always pointing along the line to the origin). Which of the following are guaranteed to be constant? (Select all that apply.)

A comet moves under the Sun's gravity with total energy $E > 0$ and angular momentum $L \ne 0$. What is the shape of its orbit?

For a particle in the plane under a central potential, $\mathcal L = \tfrac12 m(\dot r^2 + r^2\dot\theta^2) - V(r)$. What is the Euler–Lagrange equation for $\theta$?

In Lean, r v : ℝ → Fin 3 → ℝ describe a motion with hr : ∀ t, HasDerivAt r (v t) t. Which extra hypothesis expresses “the force is central”, so that r t ⨯₃ v t is constant?

Write verlet(x0, v0, h, n, accel) implementing the velocity Verlet method for $x'' = a(x)$: each step does $v \leftarrow v + \tfrac h2 a(x)$, then $x \leftarrow x + h v$, then $v \leftarrow v + \tfrac h2 a(x)$. Return the final (x, v). Test it on the harmonic oscillator $a(x) = -x$: after one period $2\pi$ the state should return to its start, and the energy $\tfrac12 v^2 + \tfrac12 x^2$ should stay almost constant.