# 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 ${\displaystyle {\bar {v}}}$.

### 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

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

implies

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

## 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.

${\displaystyle v(t)=\lim _{\Delta t\to 0}{\frac {\Delta x}{\Delta t}}=\lim _{\Delta t\to 0}{\frac {x(t_{f})-x(t_{i})}{t_{f}-t_{i}}}}$

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

${\displaystyle v(t)=\lim _{\Delta t\to 0}{\frac {x(t+\Delta t)-x(t)}{\Delta t}}={\frac {dx}{dt}}}$

### CALCULUS: Problem involving velocity, acceleration, and equations of motion

A particle's motion[1] is described by the equation ${\displaystyle \ x(t)=2t^{3}+5t+2}$. 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.