13.4 Velocity & Other Rates of Change

Annecy, French Alps

2

Slopes of Displacement FunctionsThe slope of a function is equal to the rate of change of that function:

dx

dySlope = rate of change =

Let f(t) be a function that describes how far an object is from its starting point (displacement)

ntdisplaceme)( tfs

dt

ds)(' tf

t

tfttft

0

lim

Time

Distance

VelocityousInstantane

3

Velocity versus Speed• In Calculus velocity and speed are not the same thing.

t

tfttf

dt

dstv

t

0

lim)(

Instantaneous Velocity:

Speed:

dt

dstv )(Speed

Speed does not include direction.

Velocity includes direction.

4

Velocity in Graphs

M = Average Speed

)(tvdt

ds

S(t)

5

Freefall Constants• On the planet earth, the gravitational force is constant.

In feet:

G= 32 feet per second per second

2s

ft32g

In meters

G= 9.8 meters per second per second

2s

m8.9g

6

Freefall on Earth• The formulas for freefall are shown below. They are used to

calculate the position (s) of any object at any given time (t).

In feet:

00232

2

1)( stvtts

In meters

0028.9

2

1)( stvtts

00216)( stvtts 00

29.4)( stvtts

Initial Velocity

Initial Position

7

Freefall and Derivatives• Using the formula for freefall, formulas for

velocity and acceleration can be quickly computed.

00216)( stvtts

032 vtdt

ds velocity)( tv

322

2

dt

sdvelocityinchange

dt

dv

onaccelerati)( ta

8

Freefall and Derivatives• A penny is dropped off the top of the Eiffel Tower (972

ft). When it hits the ground, how fast is it traveling?

9

Freefall and Derivatives• A penny is thrown straight up off the top of the Eiffel

Tower (972 ft). It initial speed is 64 feet per second. When it hits the ground, how fast is it traveling?

10

Freefall and Derivatives• A penny is thrown straight down off the top of the Eiffel

Tower (972 ft). It initial speed is 64 feet per second. When it hits the ground, how fast is it traveling?

11

A person on ground level throws an object upwards such that the height, h, in feet at time t, in seconds, is given by h(t) = -16t2 + 68t.

a. Find the velocity and acceleration.b. Determine the displacement over the interval

[1, 4].c. Determine the average velocity over the

interval [1, 4].d. Determine when the object hits the ground.e. Find the velocity and acceleration of the object

at the time it hits the ground.

12

Particle Motion Problems• Atomic particle are subject to many different forces besides

gravity. Therefore, the motion of a particle can take a multitude of forms. To limit the possibilities, we will only deal with particles that travel forward & backward on a line.

Example:

A particle is moving along a line. The position of the particle can be described by the equation:

08)( 24 tttts

Describe the motion of the particle.

13

Particle Motion ProblemsA particle is moving along a line. The position of the particle can be described by the equation:

08)( 24 ttttsDescribe the motion of the particle.

Here is a graph of s(t):

14

Particle Motion ProblemsA particle is moving along a line. The position of the particle can be described by the equation:

08)( 24 tttts

tttv 164)( 3

15

Particle Motion ProblemsA particle is moving along a line. The position of the particle can be described by the equation:

08)( 24 tttts

tttv 164)( 3 1612)( 2 tta

16

Particle Motion ProblemsA graph of a particle velocity is shown.

1.) What does the graph of position look like?

2.) What does the graph of acceleration look like?

17

Particle Motion ProblemsA graph of a particle velocity is shown.

1.) What does the graph of position look like?

2.) What does the graph of acceleration look like?

18

Particle Motion ProblemsA graph of a particle acceleration is shown.

1.) What does the graph of velocity look like?

2.) What does the graph of position look like?

19

A particle moves along the x-axis and its positionat time t is given by s(t) = t3 – 12t2 + 27t, where t is measured in seconds and s in feet.

a. Find the velocity and acceleration.b. Determine the velocity and acceleration after 5

seconds.c. Determine when the particle is at rest.d. Determine when the particle is moving forward.e. Find the displacement of the particle during the first 6

seconds.f. Determine the velocity of the particle when there is no

acceleration.

0t

20

A particle moves along the x-axis and its positionat time t is given by , where t is measured in seconds and s in feet.

a. Find the velocity and acceleration.b. Determine the velocity and acceleration after 5

seconds.c. Determine when the particle is at rest.d. Determine when the particle is moving forward.e. Find the displacement of the particle during the first 6

seconds.f. Determine the acceleration of the particle when

velocity is at a minimum.

2)2)(3()( ttts 0t