We now want to learn how FAR it travels in any TIME.

When an object�ACCELERATES UNIFORMLY,�we know how FAST�it is traveling at any TIME.

DISTANCE as a function of TIME

I can . . .

• apply velocity and acceleration to determine the distance an object moves as a function of time.
• apply the appropriate kinematic equation to solve a problem.
• explain the meaning of the area under the curve of a graph involving position, velocity, or acceleration with respect to time.
• do the math!!!

http://static.howstuffworks.com/gif/speedometer-1.jpg

meters per second

The velocity of an object in freefall changes by approximately�10 m/s each second.

An object in freefall accelerates at approximately�10 m/s²

http://static.howstuffworks.com/gif/speedometer-1.jpg

meters per second

The velocity of an object in freefall changes by approximately�10 m/s each second.

An object in freefall accelerates at approximately�10 m/s²

If and object went 10 m/s for the whole first second, it would have traveled 10 meters in that second.

The object actually started at zero m/s�and accelerated up to 10 m/s,�but it only AVERAGED 5 m/s.

It only traveled 5 meters during the first second.

d = 10 m/s · 1 s

d = 10 meters

d = 5 meters

m/s

m/s

m/s

s

s

s

s

s

An object in free fall travels FARTHER each second�because it is moving FASTER each second.

How far will you travel if you�AVERAGE 50 m/s for 3 s?

150 m

Average Velocity for�Constant Acceleration

What was your average velocity if you accelerate from 20 m/s to 30 m/s?

25 m/s

Distance Traveled for:

If dropped from rest, V₀ = 0.

Average Velocity for�Constant Acceleration

What was your average velocity if you accelerate from REST to 30 m/s?

15 m/s

meters per second

Acceleration = 3 m/s²

?

Time = seconds

0

1

2

3

4

5

6

7

8

10

9

How far did the object travel during the�10 seconds?

d = 15 · 10

d = 150 m

meters per second

Acceleration = 4 m/s²

?

Time = seconds

0

1

2

3

4

5

How far did the object travel during the�5 seconds?

d = 10 · 5

d = 50 m

Car Accelerates a = 8 m/s²

8 m/s

16 m/s

24 m/s

32 m/s

40 m/s

4 m

12 m

20 m

28 m

36 m

4 m

16 m

36 m

64 m

100 m

0 s

1 s

2 s

3 s

4 s

5 s

0 m/s

8 m/s

16 m/s

24 m/s

32 m/s

40 m/s

4 m/s

12 m/s

20 m/s

28 m/s

36 m/s

Ball is dropped g ≈ 10 m/s²

10 m/s

20 m/s

30 m/s

40 m/s

50 m/s

5 m

15 m

25 m

35 m

45 m

5 m

20 m

45 m

80 m

125 m

30 m/s

20 m/s

10 m/s

0 m/s

40 m/s

80 m

45 m

20 m

5 m

3 sec

2 sec

1 sec

0 sec

4 sec

1 sec

2 sec

3 sec

4 sec

If you throw a ball up at 40 m/s,

How long will it be in the air?

How high will it go?

If a ball is thrown so it is in the air�for 6 seconds,

How fast was it thrown?

How high will it go?

8 seconds

80 meters

30 m/s

45 meters

a·t

d = · t

d = ½ a · t²

V_f = a·t

d = ½·a·t²

d = ½·g·t²

(Any Acceleration)

(Acceleration of GRAVITY)

g = 9.80 m/s² or g = 32.2 ft/s²

g = 10 m/s²

d = ½ a · t²

d = ½ 10 · 1²

d = ½ 10 · 2²

d = ½ 10 · 3²

d = ½ 10 · 4²

d = ½ 10 · 5²

El Capitan, Yosemite National Park

www.nolimitstahoe.com/ adventures/photos.htm

Mike Corbit

Meters d = ½(9.80)·t²

Feet d = ½(32.2)·t²

El Capitan is 3,000 ft tall

3,000 = ½(32.2)·t²

= 14 s

Royal Gorge

8.0 seconds

​

Feet d = ½(a)·t²

d = ½(32.2)·8.0²

d = 1030.4 feet

Lyons Ferry Bridge

165 = ½(32.2)·t²

= 3.20 s

Feet d = ½·a·t²

V_f = a · t

V_f = 32.2 · 3.20

seconds

0

1

2

3

4

5

6

7

8

9

Time

V_f = a · t

Feet d = ½·a·t²

d = ½(32.2)·9.0²

d = 1300 ft

V_f = 288

· ·

1 mi�5280 ft

3600 s�1 hr

= 2.0 x 10²

m

0 = 70.0 + 12.0·t + 1/2(-9.8)t²

y

-70.0 = 0 + 12.0·t + 1/2(-9.8)t²

-70.0

v_x = +12.0 + (-9.8)t

Time = seconds

0

1

2

3

4

5

6

7

8

10

9

V₀ = 50 m/s

Hang Time

http://upload.wikimedia.org/wikipedia/commons/d/d3/Spiegel_Building_Hamburg_3.jpg

Time = seconds

10

V₀ = 50 m/s

Hang Time

http://upload.wikimedia.org/wikipedia/commons/d/d3/Spiegel_Building_Hamburg_3.jpg

How high did the ball go?

d = ½·a·t²

d = ½·10·5²

d = 125 meters

I can . . .

• apply velocity and acceleration to determine the distance an object moves as a function of time.�
• apply the appropriate kinematic equation to solve a problem.
• explain the meaning of the area under the curve of a graph involving position, velocity, or acceleration with respect to time.
• do the math!!!

d = 50 m/s · 5 s

d = 250 m

Area under the “Curve”

x = ½·a·t²

x = ½·10·5²

x = 125 m

x = 25 m/s · 5 s

x = 125 m

x = ½ 50 m/s · 5 s

Area under the “Curve”

x = v_x0·t + ½·a·t²

x = 10·5 + ½·10·5²

x = 175 m

x = ½ (10+60) · 5

x = 175 m

x = 35m/s · 5 s

Area under the “Curve”

x = A_rectangle + A_triangle

x = 10·5 + ½·50·5

x = 175 m

20·2

½(60+20)·2

½(20+30)·1

40 m

80 m

25 m

145 m

½ (-20)·2

½(30)·3

-20 m

45 m

25 m

Find total distance traveled.

What does the area under the curve give you?

In a race, which slide will win?

The moment the sports car traveling at 20 m/s passes the police car at rest, the police car begins to accelerate at a rate of 4 m/s².

Find the time when the police car catches up to the sports car:

A) Using Graphs

B) Using formulas

The moment the sports car traveling at 20 m/s passes the police car� at rest, the police car begins to accelerate at a rate of 4 m/s².

d = v₀·t

d = ½ Δv·t

d = ½ (a·t)·t

d = ½ a·t²

d = ½ a·t² + v₀·t

Acceleration is CONSTANT.

Velocity is changing by the same amount each second.

Velocity is INCREASING each second

Distance traveled during EACH second is getting GREATER.

Distance traveled during EACH second is getting GREATER.

Area = Base · Height

= m/s (velocity)

= (acceleration)

Area = ½ Base · Height

= m (distance)

= (velocity)

(50,13)

(50, 13)

(0, -3)

y = 5 · x²

y = 5 · x + 0

y = m · x + b

d = (½g)(t²)

5 = ½g

g = 10