https://www.physik.uni-leipzig.de/jvwikis/mechanics/ 2026-05-01T06:05:21+00:00 https://www.physik.uni-leipzig.de/jvwikis/mechanics/ https://www.physik.uni-leipzig.de/jvwikis/mechanics/_media/wiki/favicon.ico text/html 2021-11-18T01:52:03+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap3/3.1_motivation_and_outline?rev=1637196723&do=diff 3.1 Motivation and outline: What is causing motion? Every now and then I make the experience that I sit in a train, reading a book. Then I look out of the window, realize that we are passing a train, feeling happy that we are further approaching my final destination; and then I realize that the train is moving and my train is still in the station. Indeed, the motion of objects in my compartment is exactly identical, no matter whether it is at rest or moves with a constant velocity; be it zero i… text/html 2021-11-18T02:03:39+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap3/3.2_time_derivatives_of_vectors?rev=1637197419&do=diff 3.2 Time derivatives of vectors In this section we consider the motion of a particle with mass $m$ that is at position $\mathbf q(t)$ at time $t$. Its average velocity ${\mathbf v}_{\text{av}}(t, \Delta t)$ during the time interval $[t, t+\Delta t]$ is \begin{align*} {\mathbf v}_{\text{av}} (t, \Delta t) = \frac{ \mathbf q(t + \Delta t) - \mathbf q(t) }{\Delta t} \end{align*} When the limit $\lim_{\Delta t \to 0} \mathbf v_\text{av}(t, \Delta t)$ exists we can define the velocity of the par… text/html 2024-02-01T00:13:49+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap3/3.3_newton_s_axioms_and_equations_of_motion_eom?rev=1706742829&do=diff 3.3 Newton's axioms and equations of motion (EOM) In Section 3.1 we referred to a train compartment to point out that physical observations will be the same --- irrespective of the velocity of its motion, as long as it is constant. A setting where we perform an experiment is denoted as reference frame, and reference frames that move with constant velocity are called inertial systems.$(\mathbf Q, \{\hat{\boldsymbol e}_i(t), \; i=1\cdots D\})$$\mathbf Q(t)$$\{\hat{\boldsymbol e}_i(t), \; i=1\cdot… text/html 2024-12-16T15:32:50+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap3/3.4_constants_of_motion_cm?rev=1734359570&do=diff 3.4 Constants of motion (CM) In the previous section we saw that Newton's laws can be expressed as equations relating the second derivative of the position of a particle to the forces acting on the particle. The forces are determined as part of setting up the physical model. Subsequently, determining the time dependence of the position is a mathematical problem. Often it can be solved by finding constraints on the solution that must hold for all times. Such a constraint is called a$\mathcal{C} … text/html 2021-11-28T12:05:52+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap3/3.5_worked_example_flight_of_an_earth-bound_rocket?rev=1638097552&do=diff 3.5 Worked example: Flight of an Earth-bound rocket In order to illustrate the applications of Newton's laws we discuss now the flight of a rocket. We will deal with the case a) where the rocket is moving in vertical direction, b) where the fuel is ejected with a constant speed $v_f$$V_R$$m$$m+M_0$$M_0$\begin{align*} F_R = \bigl( m+M(t) \bigr) \: \dot V_R = a \, \rho \, v_f^2 - \bigl( m+M(t) \bigr) \: g \end{align*}$M(t)$$t$$\dot M = -a \, \rho \, v_f$$M(0)=M_0$$\dot M(t)$\begin{align*} M(t… text/html 2022-01-30T04:23:48+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap3/3.6_problems?rev=1643513028&do=diff 3.6 Problems 3.6.1 Practicing Concepts Problem 3.13: Car on an air-cushion We consider a car of mass $m=20 \, \text{g}$ moving -- to a very good approximation without friction -- on an air-cushion track. There is a string attached to the car that moves over a roll and hangs vertically down on the side opposite to the car.$F = 2 \, \text{N}$$v(t)$$x(t)$$200 \, \text{g}$\begin{align*}E = E_{\mathrm{kin}} + E_{\mathrm{pot}} = \frac{m+M}{2} \; v^2 + Mgh = \hbox{const}\end{align*}$M$$v_I = 26 … text/html 2022-01-02T15:00:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap3/3.7_further_reading?rev=1641132013&do=diff 3.7 Further reading Sommerfeld’s (1994) classical discussion of Newton's axioms dates back to the 1940s, but still is a one of the most superb expositions of the topic. A comprehensive discussion of the flight of water bottle rockets has been given in text/html 2022-04-01T19:49:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.physik.uni-leipzig.de/jvwikis/mechanics/book/chap3/newton?rev=1648835362&do=diff 3. Newton's Laws In Chapter 2 we explored how several forces that act on a body can be subsumed into a net total force and torque. The body stays in rest, say at position $\mathbf q_0$, when the net force and torque vanish. Now we explore how the forces induce motion and how the position of the body evolves in time, $\mathbf q(t)$$\mathbf q(t_0) = \mathbf q_0$$t_0$