Antidifferentiation method for kinematics

2024-09-2711:45 Status:PHYS106 Tags: Kinematics

We can use small time steps method to make predictions: https://scratch/mit/edu/projects/576886731/editor

Antidifferentiation method

This method can be used to find $v(t)$ and $x(t)$ explicitly if $F$ is a known function of time: If force is a known function of time, $\frac{F}{m}=a(t)$, then $\frac{dv}{dt}=a(t)$. To find $v(t)$ given $v(t_0)$, find a function $w(t)$ whose derivative is $a(t)$. Then $v(t)=w(t)+C$. Choose the number $C$ so that the velocity at $t=t_0$.

If force is a known function of time, $\frac{dv}{dt}=f(t)$. To find $v(t)$ given $v(t_0$): Find a function $g(t)$ whose derivative is $f(t)$. Then $v(t)=g(t)+C$. Choose the correct number $C$ so that the velocity at $t=t_0$ is correct. Use the same procedure to find $x(t)$ given $v(t)$ and $x(t_0)$. I.e. $\frac{dx}{dt}=v(t)$.

Note: constant acceleration is when $a=\frac{F}{m}$ is constant: $\frac{dv}{dt}=a$. Thus, $\frac{dv}{dt}=a\to v(t)=v_0+at$. Moreover, $\frac{dv}{dt}=v_0+at\to x=x_0+v_{0}t+\frac{1}{2}a{t}^2$. This can be derived by remembering the relationship between $x$ and its integral, $v$ and its derivative/integral, and $a$ and its integral (or antidifferentiation). And that $z_{new}=z_{old}+\frac{dz}{dt}t$ Note that $\frac{dx}{dt}$ can be substituted for “velocity”. Which has the same general formula. This nesting substitution can go on until there is a constant rate.

Example

\\ \vec x= t^2 + t + C \\ (3) = (1)^2+(1) +C \ \ \ \ \therefore C=1 \\ at \ t=3, \vec x =(3)^2 + 3 + 1 = 13 \end{gather}$$