book:chap3:3.3_newton_s_axioms_and_equations_of_motion_eom
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| book:chap3:3.3_newton_s_axioms_and_equations_of_motion_eom [2024/02/01 00:05] – [3.3.5 Self Test] jv | book:chap3:3.3_newton_s_axioms_and_equations_of_motion_eom [2024/02/01 00:13] (current) – fixing typos in mine cart exampe jv | ||
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| Line 336: | Line 336: | ||
| \end{align*} | \end{align*} | ||
| The mine cart travels with constant velocity $\dot v = 0$, when the attacking forces balance, | The mine cart travels with constant velocity $\dot v = 0$, when the attacking forces balance, | ||
| - | i.e., for $v_c = F_M / m\, \gamma$. | + | i.e., for $v_c = F_M / \gamma$. |
| For a different initial velocity, $v(t_0) = v_0$, one finds an exponential approach to the asymptotic velocity, | For a different initial velocity, $v(t_0) = v_0$, one finds an exponential approach to the asymptotic velocity, | ||
| \begin{align*} | \begin{align*} | ||
| - | v (t) = v_c + \bigl( v_0 - v_c \bigr) \; \mathrm{e}^{ - \gamma \, (t-t_0) } | + | v (t) = v_c + \bigl( v_0 - v_c \bigr) \; \mathrm{e}^{ - \gamma \, (t-t_0) |
| \end{align*} | \end{align*} | ||
| After all, | After all, | ||
| Line 345: | Line 345: | ||
| and | and | ||
| \begin{align*} | \begin{align*} | ||
| - | \dot v (t) | + | |
| - | &= \bigl( v_0 - v_c \bigr) \; (-\gamma) \, \mathrm{e}^{ - \gamma \, (t-t_0) } | + | &= \bigl( v_0 - v_c \bigr) \; (-\gamma) \, \mathrm{e}^{ - \gamma \, (t-t_0)/m } |
| \\ | \\ | ||
| - | & | + | &= -\gamma \; \bigl( v(t) - v_c \bigr) = -\gamma \; v(t) + F_M |
| \end{align*} | \end{align*} | ||
| </ | </ | ||
book/chap3/3.3_newton_s_axioms_and_equations_of_motion_eom.1706742301.txt.gz · Last modified: 2024/02/01 00:05 by jv