MOTION

Chapter Four: Motion

Ø 4.1 Position, Speed and Velocity

Ø 4.2 Graphs of Motion

Ø 4.3 Acceleration

4.1 Position, Speed and Velocity Ø Position is a variable given relative to an

origin. Ø The origin is the place where position equals 0.

Ø The position of this car at 50 cm describes where the car is relative to the track.

4.1 Position, Speed and Velocity Ø Position and distance are similar but not

the same. Ø If the car moves a distance of 20 cm to the

right, its new position will be 70 cm from its origin.

Distance = 20 cm

New position

4.1 Position, Speed and Velocity Ø The variable speed describes how quickly

something moves. Ø To calculate the speed of a moving object

divide the distance it moves by the time it takes to move.

4.1 Position, Speed and Velocity Ø The units for speed are distance units over

time units.

Ø This table shows different units commonly used for speed.

4.1 Average speed Ø When you divide the

total distance of a trip by the time taken you get the average speed.

Ø On this driving trip around Chicago, the car traveled and average of 100 km/h.

4.1 Instantaneous speed Ø A speedometer shows

a car’s instantaneous speed.

Ø The instantaneous speed is the actual speed an object has at any moment.

How far do you go if you drive for two hours at a speed of 100 km/h?

1.   Looking for: Ø  …distance

2.   Given: Ø  …speed = 100 km/h time = 2 h

3.   Relationships: Ø  d = vt

4.   Solution: Ø  d = 100 km/h x 2 h = 200 km

= 200 km

Solving Problems

4.1 Vectors and velocity Ø  Position uses positive and negative

numbers. Ø  Positive numbers are for positions to the

right of the origin and negative numbers are for positions to the left the origin.

4.1 Vectors and velocity

Ø  Distance is either zero or a positive value.

4.1 Vectors and velocity Ø  We use the term velocity to mean

speed with direction.

4.1 Keeping track of where you are Ø  Pathfinder is a small robot sent to

explore Mars.

Ø  It landed on Mars in 1997.

Ø  Where is Pathfinder now?

4.1 Keeping track of where you are Ø  Pathfinder keeps track of its

velocity vector and uses a clock.

Ø  Suppose Pathfinder moves forward at 0.2 m/s for 10 seconds.

What is Pathfinder’s velocity?

4.1 Keeping track of where you are Ø  Suppose Pathfinder goes backward

at 0.2 m/s for 4 seconds.

What is Pathfinder’s change in position?

4.1 Keeping track of where you are Ø The change in position is the velocity multiplied by the time.

4.1 Keeping track of where you are Ø Each change in position is added up using positive and negative numbers.

Ø Pathfinder has a computer to do this.

4.1 Maps and coordinates Ø  If Pathfinder was crawling on a straight

board, it would have only two choices for direction.

Ø  Out on the surface of Mars, Pathfinder has more choices.

Ø  The possible directions include north, east, south, and west, and anything in between.

4.1 Maps and coordinates Ø  A graph using north−south and

east−west axes can accurately show where Pathfinder is.

Ø  This kind of graph is called a map.

Ø  Street maps often use letters and numbers for coordinates.

4.1 Vectors on a map Ø  Suppose you run east for 10 seconds at

a speed of 2 m/s.

Ø  Then you turn and run south at the same speed for 10 more seconds.

Ø  Where are you compared to where you started?

4.1 Vectors on a map Ø To get the answer, you figure out your east−west changes and your north−south changes separately. origin = (0 , 0)

4.1 Vectors on a map Ø Your first

movement has a velocity vector of +2 m/s, west-east (x-axis).

Ø After 10 seconds your change in position is +20 meters (east on x-axis).

d = v x t d = 2 m/s x 10 s = +20 m

4.1 Vectors on a map Ø Your second

movement has a velocity vector of −2 m/s north−south (y-axis)

Ø In 10 seconds you move −20 meters (south is negative on y-axis)

d = 2 m/s x 10 s = -20 m New position = (+20 , -20)

A train travels at 100 km/h heading east to reach a town in 4 hours. The train then reverses and heads west at 50 km/h for 4 hours. What is the train’s position now?

1.   Looking for: Ø  …train’s new position

2.   Given: Ø  …velocity = +100 km/h, east ; time = 4 h Ø  …velocity = -50 km/h, west ; time = 4 h

3.   Relationships: Ø  change in position = velocity × time

Solving Problems

4.   Solution: Ø  1st change in position: Ø  (+100 km/h) × (4 h) = +400 km

Ø  2nd change in position: Ø  (−50 km/h) × (4 h) = −200 km

Ø  Final position: Ø  (+400 km) + (−200 km) = +200 km

Ø  The train is 200 km east of where it started.

Solving Problems

4.2 Graphs of Motion Ø  Constant speed means the speed stays

the same. Ø  An object moving at a constant speed

always creates a position vs. time graph that is a straight line.

4.2 Graphs of Motion Ø  The data shows the

runner took 10 seconds to run each 50-meter segment.

Ø  Because the time was the same for each segment, you know the speed was the same for each segment.

4.2 Graphs of Motion Ø  You can use

position vs. time graphs to compare the motion of different objects.

Ø  The steeper line on a position vs. time graph means a faster speed.

4.2 Slope Ø  The steepness of a line is measured

by finding its slope. Ø  The slope of a line is the ratio of the “rise” to the “run”.

4.2 Graphs of changing motion Ø  Objects rarely move

at the same speed for a long period of time.

Ø  A speed vs. time graph is also useful for showing the motion of an object that is speeding up or slowing down.

4.2 Graphs of changing motion Ø  Suppose we draw a rectangle on the

speed vs. time graph between the x-axis and the line showing the speed.

Ø  The area of the rectangle is equal to its length times its height.

Ø  On the graph, the length is equal to the time and the height is equal to the speed.

4.3 Acceleration Ø Acceleration is the rate at which your

speed (or velocity) changes. Ø If your speed increases by 1 meter per

second (m/s) for each second, then your acceleration is 1 m/s per second.

4.3 Acceleration Ø Acceleration is easy to spot on a speed vs. time graph.

Ø Acceleration causes the line to slope up on a speed vs. time graph. What is the bike’s acceleration?

4.3 Acceleration Ø If the hill is steeper, the acceleration

is greater.

4.3 Acceleration Ø There is zero acceleration at constant

speed because the speed does not change.

4.3 Acceleration Ø Speed and acceleration are

not the same thing. Ø You can be moving (non-

zero speed) and have no acceleration (think cruise control).

Ø You can also be accelerating and not moving!

Ø A falling object begins accelerating the instant it is released.

4.3 Acceleration Ø Acceleration describes how quickly speed changes.

Ø Acceleration is the change in speed divided by the change in time.

4.3 Speed and acceleration Ø An acceleration of 20

km/h/s means that the speed increases by 20 km/h each second.

Ø The units for time in acceleration are often expressed as “seconds squared” and written as s2.

Can you convert this rate using conversion factors?

Solving Problems

Ø A sailboat moves at 1 m/s.

Ø A strong wind increases its speed to 4 m/s in 3 s.

Ø Calculate acceleration.

1.   Looking for:

Ø  …acceleration of sailboat

2.   Given:

Ø  …v1 = 1 m/s; v2 = 4 m/s; time = 3 s

3.   Relationships:

Ø  a = v2 – v1/t

4.   Solution:

Ø  a = (4 m/s – 1 m/s)/ 3 s = 1 m/s2

Solving Problems

4.3 Acceleration on motion graphs Ø The word “acceleration” is used for any change in speed, up or down.

Ø Acceleration can be positive or negative.

4.3 Acceleration on speed-time graphs

Ø Positive acceleration adds more speed each second.

Ø Things get faster.

Ø Speed increases over time.

4.3 Acceleration on speed-time graphs

Ø Negative acceleration subtracts some speed each second.

Ø Things get slower.

Ø People sometimes use the word deceleration to describe slowing down.

4.3 Acceleration on position-time graphs

Ø The position vs. time graph is a curve when there is acceleration.

Ø The car covers more distance each second, so the position vs. time graph gets steeper each second.

4.3 Acceleration on position-time graphs

Ø When a car is slowing down, the speed decreases so the car covers less distance each second.

Ø The position vs. time graph gets shallower with time.

4.3 Free fall Ø An object is in

free fall if it is accelerating due to the force of gravity and no other forces are acting on it.

4.3 Free fall Ø Falling objects increase their speed by 9.8 m/s every second, or 9.8 m/s2

Ø The letter “g” is used for acceleration due to gravity.

4.3 Acceleration and direction Ø  Acceleration occurs whenever there

is a change in speed, direction, or both.

4.3 Acceleration and direction Ø  A car driving around a curve at a

constant speed is accelerating because its direction is changing.

4.3 Acceleration and direction Ø  Individual vectors can be drawn to

scale to calculate the change in direction.

4.3 Curved motion Ø A soccer ball is an

example of a projectile.

Ø A projectile is an object moving under the influence of only gravity.

Ø The path of the ball makes a bowl-shaped curve called a parabola.

4.3 Curved motion

Ø Circular motion is another type of curved motion.

Ø An object in circular motion has a velocity vector that constantly changes direction.