Kinematics – the study of how things move

When an object moves, what changes?•position

•time

The rate at which an object moves is called its average speed.

Average Speed (v) – distance traveled / time interval

v = d / t

Kinematics

Graphing Position and Time

Position (m) Time (s)

0 0

1 1

4 2

7 3

8 4

9 5

12 6

Pos

ition

(m

)

Time (s)

P/T Graph

Best-fit line tells you the average speed. NOT required to go through any of the points

Distance and speed are scalar quantities. Scalar quantities are always positive. Direction is not significant.

In kinematics, direction sometimes matters.

Displacement – a change in position

Δ x = xf – xi

Δ x = change in positionxf = final positionxi = initial position

Average Velocity – change in position / change in time

v = Δ x /Δ t = xf – xi / tf – tiTypically, the positive direction is:

To the right, upward, to the east, northward

The negative direction is:To the left, downward, to the west, southward

Displacement and Velocity are vector quantities because direction matters.

A runner changes his position from x = 30.5 m to x = 50.0 m in 3.00 seconds. What is the runner’s average speed?

He immediately returns to his starting position in 4.50 seconds. What is his average speed over the entire trip?

What is his average velocity?

Graphing Position and Time

Position (m) Time (s)

0 0

2 1

4 2

6 3

6 4

3 5

0 6

Pos

ition

(m

)

Time (s)Notice the positive and negative velocities.

P/T Graph

Graphing Velocity and Time

Velocity (m/s) Time (s)

2 0

2 1

2 2

2 3

0 4

-3 5

-3 6

Vel

ocity

(m

/s)

Time (s)Displacement is the area(s) under the line

V/T Graph

Acceleration

Position (m) Time (s)

0 0

1 1

3 2

6 3

10 4

15 5

21 6

Pos

ition

(m

)

Time (s)

P/T Graph

Average Acceleration – change in velocity / change in time

a = Δ v /Δ t = vf – vi / tf – ti

Every time on the graph has a unique instantaneous velocity. The relationship between the velocities is:

If an object has an acceleration of ___ m/s2, every second its velocity increases ______ m/s.

A car accelerates along a straight road from rest to 75 m/s in 5.0 seconds. What is its average acceleration?

What is the car’s velocity after 3.0 seconds? After 8.0 seconds?

Kinematic Equations

slope = acceleration

a = Δv / Δt

a = vf – vi / Δt

aΔt = vf - vi

vf = vi + at

The kinematic equations are used to study the motion of accelerating objects. Which equation is used depends on what you are given and what you are trying to find.

“How fast?”

“How far?”

area = displacement

Δx = area A + area B

Δx = ½ bh + bh

Δx = ½(vf - vi)Δt + viΔt

Δx = ½vfΔt – ½viΔt + viΔt

Δx = ½vfΔt + ½viΔt

Δx = ½(vi + vf)Δt

• A sidewinder missile is dropped from a fighter jet traveling 200 m/s. If the missile immediately attains an acceleration of 15 m/s2, how fast is it going after 10 seconds?

A diver drops off of a cliff with an unknown height and falls toward the water. If he hits the water 3.40 seconds later traveling a velocity of 35 m/s, how high is the cliff?

Don’t know “vf”?

Δx = ½(vi + vf)Δt

Δx = ½(vi + vi + aΔt)Δt

Δx = ½(2vi + aΔt)Δt

Δx = (vi + ½aΔt)Δt

Δx = vit + ½at2

vf = vi + aΔt

Don’t know “Δt”?

Δx = ½(vi + vf)Δt

Δx = ½(vi + vf) (vf – vi) / a

aΔx = ½(vi + vf) (vf – vi)

2aΔx = (vi + vf) (vf – vi)

2aΔx = vf2 - vi

2

vf2 = vi

2 + 2aΔx

vf = vi + aΔt

vf - vi = aΔt

Δt = (vf – vi) / a

An airplane must reach a speed of at least 27.8 m/s in order to takeoff. It can accelerate at 2.00 m/s2. If the runway is 150 m long, can this airplane reach the required velocity for takeoff? If not, what minimum length must the runway have?

Freefall Problems

When an object is falling in space without any forces acting on it except gravity, it is considered a “freefall” problem. To solve freefall problems, some things to remember include:

•Air resistance is negligible unless otherwise stated

•Objects that return to their starting position have a displacement of “0”

•Objects at their maximum height have a velocity of “0”

•Gravity is always acting on freefall objects with an acceleration of 9.8 m/s2 (g)

Suppose a ball is dropped from a tower 70.0 m high. How long would it take for the ball to hit the ground?

A stone is thrown from a cliff straight down with a speed of 3.00 m/s. If it hits the ground 4.00 seconds after it is thrown, how high is the cliff?

A person throws a ball upward into the air with an initial velocity of 15.0 m/s. Calculate how high it goes and how long the ball is in the air before it comes back to his hand.

Designate upward direction as positive and use –g for acceleration because it is directed downward.

Projectile Motion

A projectile is an object in freefall that is moving in two dimensions. Resultant trajectory is parabolic because:

Horizontal motion is a constant velocity situation (no air resistance)

Vertical motion is a changing velocity/constant acceleration situation (due to gravity)

Because of the different situations, parabolic motion has to be broken down into x- and y- directions.

A movie stunt driver on a motorcycle speeds horizontally off a 50.0-m high cliff. How fast must the motorcycle leave the cliff top to land on level ground below, 90.0 m from the base of the cliff?

A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff and 3.0 s later reaches the water below. How high was the cliff, and how far from its base did the diver hit the water?

Horizontal and vertical motion act simultaneously but also independently.

The horizontal component of the yellow ball’s motion does not affect the time it takes for it to move downward. Both balls would strike the floor simultaneously.

Projectiles at an Angle

The angle at which a projectile moves is found by combining the horizontal and vertical components of its velocity.

Move vectors so that they are tail-to-tip (vector addition)

If they form a right angle, hypotenuse is the resultant velocity.

Use right triangle equations to find magnitude and angle.

You observe the resultant velocity, but you analyze the components.

If two vectors are NOT perpendicular to each other and you want to add them together

1. Break each vector into x and y components

2. Add x components together and y components together

3. Combine x vector and y vector to get resultant vector

A football is kicked at an angle of 37.0 degrees with a velocity of 20.0 m/s. Calculate the maximum height and how far away it hits the ground.

A projectile is launched from ground level to the top of a cliff which is 195 m away and 155 m high. If the projectile lands on top of the cliff 7.6 s after it is fired, find the initial velocity of the projectile (magnitude and direction).