# Physics equations/02-One dimensional kinematics/A:practice

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### Highway exits

A driver gets on mile 25 of a highway at 3:30 pm and exits at mile 150 at 5:30 pm. If the road is straight, what is the velocity and is it average or instantaneous?

### Finding average velocity

A particle starting 23 m from the origin has moved to 43 m in 5 s. Find ${\bar {v}}$ .

### Finding the average acceleration if the direction reverses in the time interval

A person is jogging east at 3m/s when he suddenly reverses direction and is jogging west at 3m/s, taking one second to accomplish this reversal. Take east to be the 'positive' direction. What is the average acceleration?

### Proving one of the equations of motion

Use algebra to show that

$v=v_{0}+at$ and  $x=x_{0}+v_{0}t+{\tfrac {1}{2}}at^{2}$ implies

$v^{2}=v_{0}^{2}+2a\left(x-x_{0}\right)$ and  $x-x_{0}={\frac {v_{0}+v}{2}}t$ .

## CALCULUS: Motion and the mathematical definition of derivative as a limit

By definition, velocity involves two different positions at two different times. However, we may take the limit that these differences are very small and define the instantaneous velocity.

$v(t)=\lim _{\Delta t\to 0}{\frac {\Delta x}{\Delta t}}=\lim _{\Delta t\to 0}{\frac {x(t_{f})-x(t_{i})}{t_{f}-t_{i}}}$ A connection to differential calculus is seen by rewriting $t_{i}$ and $t_{f}$ as $t$ and $t+\Delta t$ , so that $x_{f}=x(t+\Delta t)$ :

$v(t)=\lim _{\Delta t\to 0}{\frac {x(t+\Delta t)-x(t)}{\Delta t}}={\frac {dx}{dt}}$ ### CALCULUS: Problem involving velocity, acceleration, and equations of motion

A particle's motion is described by the equation $\ x(t)=2t^{3}+5t+2$ . Find

a) the particle's velocity function,

b) its instantaneous velocity at t = 2 s. Also find

c) the particle's acceleration function and

d) its instantaneous acceleration at t = 2 s.