The Lumenary · Early undergraduate

Classical mechanics

Name: ______________________

Date: ______________________

You will learn to

Write Newton's laws as differential equations and solve the motion of projectiles and of harmonic oscillators, with and without damping and forcing
Prove conservation of energy for conservative forces and of momentum and angular momentum, and use them to reduce problems to one dimension
Derive Kepler's three laws from the inverse-square law of gravitation, including the conic-section orbit and the relation between energy and eccentricity
Derive the Euler–Lagrange equations from the principle of stationary action and use a Lagrangian to obtain equations of motion
Distinguish mass from weight, and explain why no force is needed to keep a body moving at constant velocity

1.
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?
2.
An astronaut has mass $70$ kg. On the Moon ( $g \approx 1.6\ \mathrm{m/s^2}$ ), which statement is correct?
- A.Her mass is $70$ kg and her weight is about $112$ N.
- B.Her mass is about $11.4$ kg and her weight is $70$ N.
- C.Her mass and her weight are both about $11.4$ kg.
- D.Her mass is $70$ kg and her weight is $686$ N.
3.
A hockey puck slides in a straight line at constant speed across perfectly frictionless ice. What is the net horizontal force on it?
- A.Zero
- B.A forward force equal to the force of the original hit
- C.A small forward force that keeps it moving
- D.It depends on the puck's mass
4.
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?
5.
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?
6.
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.The angular momentum $\mathbf L = m\,\mathbf r \times \mathbf v$
- B.The rate $dA/dt$ at which area is swept out
- C.The linear momentum $m\mathbf v$
- D.The speed $|\mathbf v|$
- E.The total energy
7.
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?
- A.A branch of a hyperbola
- B.An ellipse
- C.A parabola
- D.A straight line through the Sun
8.
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 $\theta$ ?
- A. $\dfrac{d}{dt}\big(m r^2\theta'\big) = 0$
- B. $m r^2\theta'' = 0$
- C. $m r\theta'^2 = V'(r)$
- D. $\theta'' = -\dfrac{V'(r)}{m r}$
9.
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?
- A.hv : ∀ t, HasDerivAt v (k t • r t) t for some k : ℝ → ℝ
- B.hv : ∀ t, HasDerivAt v (k t • v t) t for some k : ℝ → ℝ
- C.hv : ∀ t, HasDerivAt v 0 t
- D.hv : ∀ t, r t ⬝ᵥ v t = 0
10.
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.
Computational — Practice

Answer key — Classical mechanics

1.
$R = \dfrac{v_0^2\sin 2\alpha}{g} = \dfrac{400 \cdot \sin 90^\circ}{10} = 40$ m. Equivalently: flight time $2 \cdot 20 \cdot \tfrac{\sqrt2}{2}/10 = 2\sqrt2$ s, times horizontal speed $10\sqrt2$ m/s.
2.
Mass is unchanged: $70$ kg. Weight is $W = mg \approx 70 \times 1.6 = 112$ N on the Moon, versus about $686$ N on Earth.
3.
The velocity is constant, so $\mathbf a = 0$ and the net force is $\mathbf F = m\mathbf a = 0$ . On real ice, friction would provide a small backward force, slowing the puck.
4.
$\omega = \sqrt{k/m} = \sqrt{18/0.5} = \sqrt{36} = 6$ rad/s; the period is $2\pi/6 \approx 1.05$ s, whatever the amplitude. (Gravity only shifts the equilibrium position; it does not change $\omega$ .)
5.
Since $T^2/a^3 = 4\pi^2/GM_\odot$ is the same for every orbit around the Sun, in units of years and AU we have $T^2 = a^3 = 64$ , so $T = 8$ years.
6.
(a) and (b). The angular momentum is conserved for every central force, and the areal velocity is $L/2m$ . Momentum and speed change. Energy is conserved when $\mathbf F = f(r)\hat{\mathbf r}$ with $f$ depending only on $r$ (then $V(r) = -\int f$ ), but not for a general time-dependent central force.
7.
With $E > 0$ the eccentricity $e = \sqrt{1 + 2EL^2/(mk^2)}$ exceeds $1$ , so the orbit $r = p/(1 + e\cos\theta)$ is a branch of a hyperbola: $r \to \infty$ as $1 + e\cos\theta \to 0^+$ . The comet visits once and leaves.
8.
$\partial\mathcal L/\partial\dot\theta = mr^2\dot\theta$ and $\partial\mathcal L/\partial\theta = 0$ , so $\frac{d}{dt}(mr^2\theta') = 0$ : conservation of angular momentum, here coming from the rotational symmetry of $\mathcal L$ (Noether).
9.
Option (a): HasDerivAt v (k t • r t) t says $\mathbf r'' = k(t)\,\mathbf r$ , i.e. the force is parallel to $\mathbf r$ . The lesson proves r s ⨯₃ v s = r t ⨯₃ v t from it by differentiating each component of the cross product.
10.
```
def verlet(x0, v0, h, n, accel):
    x, v = x0, v0
    for _ in range(n):
        v = v + h / 2 * accel(x)
        x = x + h * v
        v = v + h / 2 * accel(x)
    return x, v
```
Velocity Verlet is second order and symplectic: it exactly conserves a slightly perturbed energy, so the true energy stays within a narrow band for very long times instead of drifting, which is why it is the workhorse of orbital and molecular simulations.