Velocity & Acceleration

average velocity

represented by the equation of vector v avg = vector r - vector r0 / t - t0

• thus vector v avg = vector ∆r / ∆t

  • vector ∆r is the displacement

  • ∆t is the change in time or the time interval

• average velocity describes the change in position

velocity is the the vector

parallel to displacement telling how quickly it occurred and the direction

• independent of position

average speed

average speed is equal to distance / elapsed time

• there is no information about the direction of motion

instantaneous velocity is the

instantaneous rate at which position changes (a more accurate description of motion rather than that of average velocity)

• represented by the equation vector v avg = lim ∆t → 0 vector ∆r / ∆t

• small ∆t value can be thought of as an instant

instantaneous velocity compares

intermediate positions

• the average slope of the position vs. time graph shows the average rate of change of x with time (average velocity)

• the slope of the tangent to the position vs. time graph at a specific point and time interval gives the instantaneous velocity

motion with constant velocity

if the vector v (velocity) is constant, the position of the object changes at a constant rate

• mathematically, displacement is proportional to elapsed time

• represented by the equation x(t) = x0 + vt (rearranged from the average velocity equation v = x - x0 / t - t0, when t0 = 0

velocity describes the

instantaneous direction of motion

acceleration is the

rate at which velocity changes (rate of change of velocity)

• average acceleration is represented by the equation vector a avg = vector v - vector v0 / t - t0 = vector ∆v / ∆t

  • vector ∆v is the change in velocity

  • ∆t is the change in time

• average acceleration is always parallel ot the change in velocity (vector ∆v)

acceleration has

different dimensions than velocity and is on a different scale

instantaneous acceleration is a

more accurate description of the change in velocity

• represented by the equation vector a = lim ∆t → 0 vector ∆v / ∆t

  • acceleration vector tells us the rate and direction of velocity change

  • velocity vector can change in magnitude only, direction only, or both (results in acceleration change)

acceleration may correspond to either an

increase or decrease in speed

if acceleration is in the same direction as velocity, the

speed increases

• velocity will have a greater magnitude

• either positive or negative acceleration

• vector v initial + vector ∆v = vector v final

if acceleration is in the opposite direction as velocity, the

speed decreases

• velocity will have a smaller magnitude

• either positive or negative acceleration

• vector v initial - vector ∆v = vector v final

negative acceleration does not mean

decreasing speed

if the acceleration vector is constant, then

velocity changes at a constant rate

• one dimensional

• if t0 = 0, then the equation is v(t) = v0 + at (velocity as a function of time)

  • rearranged from the equation a = v(t) - v0 / t - t0, where t0 = 0

if acceleration or velocity is constant, than the

average acceleration or velocity is equal to the instantaneous acceleration or velocity (linear function)

displacement is somewhat proportional to the

average velocity

• an object will travel over a greater distance if it moves faster

• represented by the equation x(t) = x0 + v(avg) t

  • where v(avg) = ½ (v0 + vt)

if the velocity vt is such as vt = v0 + at, by

substituting average velocity, the equation is: xt = x0 + [½ (v0 + vt)] t

• by substituting vt, this gives the equation: xt = x0 + [½ (v0 + v0 + at)] t

  • this rearranges algebraically to the equation: xt = x0 + v0 t + ½ at² (position as a function of time equation)

the equation xt = x0 + v0 t + ½ at² explains

how the position of an object depends on initial velocity (v0) and acceleration (a), when acceleration is constant

• v0 t = effect of initial velocity (linear function)

• ½ at² = effect of acceleration (quadratic function)

combining the equations for vt and xt gives the equation

v² = v0² + 2a (x - x0)

• if we set the origin at the initial point (x0 = 0), it is: v² = v0² + 2ax

• this equation directly relates an object’s velocity with the position at which that velocity is attained (since the variable time is eliminated, and there is only information about the distance)

  • distance is proportional to ∆v²