SECT. 3-APOSITION, VELOCITY, AND ACCELERATION

Position function - gives the location of an object at time t, usually s(t), x(t) or y(t)

Velocity - The rate of change( derivative) of position, usually v(t)

Acceleration -The rate of change ( derivative) of velocity usually a(t)

Average velocity -

x(t)

v(t) x (t)

a(t) v (t) x (t)

ab

afbf

time

)()(distance

1) If

find v(t) and a(t).

tttts 53)( 23

2) The position of a particle moving along the x - axis at time t is given by

. Find the particles velocity and acceleration at t = 5.

x(t)t 3 11t 2 24t

Velocity Acceleration

Speed – The absolute value of velocity otherwise known as the magnitude of velocity

)()(' tvtxspeed

Velocity

Positive - the particle is moving to the right

Negative - the particle is moving to the left

Zero - the particle has momentarily stopped or is changing direction ( must have a sign change)

3) Find where the object changes direction if

x(t)t 3 12t 2 36t 18

Find where v(t) = 0

interval

time

V(t)

Result

Acceleration and Velocity• If the sign of acceleration is the same as velocity, the

speed of the particle is ______________ (the two are working together)

• If the sign of the acceleration is opposite that of velocity, the speed of the particle is _________________ (the two are working against each other)

4) Given the same position function as #3

find the interval during which the particle is slowing down.

183612)( 23 ttttx

Interval

t

velocity

acceleration

result

5) Given find the interval during which the particle is speeding up.

360212)( 23 ttttx

Interval

t

velocity

acceleration

result

Distance vs. Displacement

•Displacement- change in position ( final position minus original position)

•Distance- the total distance travelled by an object in the time interval even if duplicated

6) Find the DISTANCE traveled by the particle whose position is given by

on the interval (0,4).

24 8)( tttx

tttv 164)( 3

Distance NOT displacement!!

00)4(4 2 tt

2,2,0 t Consider twointervals

)2()4()0()2( xxxxD

)16(1280)16( 160

7) If

a)find the DISTANCE traveled by the particle on the interval (2,4).

b)Find the DISPLACEMENT on the interval (1,5)

tttx 6)( 2

8) The graph shows the position function of a radio controlled car

a) Was the car going faster at B or at C?

b) When was the car stopped?

c) At which point was the car’s velocity

the greatest?

d) At which point was the car’s speed decreasing?

Homework

PVA Worksheet and

worksheet 3-A