MOTION

Chapter Four: Motion 4.1 Position, Speed and Velocity 4.2 Graphs of Motion 4.3 Acceleration

Section 4.1 Learning Goals Explain the meaning of motion. Describe an object’s position relative to a reference point. Use the speed formula. Tell the difference between speed and velocity.

4.1 Position, Speed and Velocity The origin is the place where position equals 0. The position of this car at 50 cm describes where the car is relative to the track. Position is a variable given relative to an origin.

4.1 Position, Speed and Velocity Position and distance are similar but not the same. 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 The variable speed describes how quickly something moves. 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 The units for speed are distance units over time units. This table shows different units commonly used for speed.

4.1 Average speed When you divide the total distance of a trip by the time taken you get the average speed. On this driving trip around Chicago, the car traveled and average of 100 km/h.

4.1 Instantaneous speed A speedometer shows a car’s instantaneous speed. The instantaneous speed is the actual speed an object has at any moment.

Solving Problems How far do you go if you drive for two hours at a speed of 100 km/h? Looking for: …distance Given: …speed = 100 km/h time = 2 h Relationships: d = vt Solution: d = 100 km/h x 2 h = 200 km = 200 km

4.1 Vectors and velocity Position uses positive and negative numbers. 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 Distance is either zero or a positive value.

4.1 Vectors and velocity We use the term velocity to mean speed with direction.

4.1 Keeping track of where you are Pathfinder is a small robot sent to explore Mars. It landed on Mars in 1997. Where is Pathfinder now?

4.1 Keeping track of where you are Pathfinder keeps track of its velocity vector and uses a clock. 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 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 The change in position is the velocity multiplied by the time.

4.1 Keeping track of where you are Each change in position is added up using positive and negative numbers. Pathfinder has a computer to do this.

4.1 Maps and coordinates If Pathfinder was crawling on a straight board, it would have only two choices for direction. Out on the surface of Mars, Pathfinder has more choices. The possible directions include north, east, south, and west, and anything in between.

4.1 Maps and coordinates A graph using north−south and east−west axes can accurately show where Pathfinder is. This kind of graph is called a map. Street maps often use letters and numbers for coordinates.

4.1 Vectors on a map Where are you compared to where you started? Suppose you run east for 10 seconds at a speed of 2 m/s. Then you turn and run south at the same speed for 10 more seconds. Where are you compared to where you started?

4.1 Vectors on a map 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 Your first movement has a velocity vector of +2 m/s, west-east (x-axis). 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 Your second movement has a velocity vector of −2 m/s north−south (y-axis) 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)

Solving Problems 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? Looking for: …train’s new position Given: …velocity = +100 km/h, east ; time = 4 h …velocity = -50 km/h, west ; time = 4 h Relationships: change in position = velocity × time

Solution: Solving Problems 1st change in position: (+100 km/h) × (4 h) = +400 km 2nd change in position: (−50 km/h) × (4 h) = −200 km Final position: (+400 km) + (−200 km) = +200 km The train is 200 km east of where it started.

Chapter Four: Motion 4.1 Position, Speed and Velocity 4.2 Graphs of Motion 4.3 Acceleration

Section 4.2 Learning Goals Construct and analyze graphs of position versus time, and speed versus time. Recognize and explain how the slope of a line describes the motion of an object. Explain the meaning of constant speed.

Constant Speed Investigation 4A Key Question: What do position vs. time and speed vs. time graphs look like for constant speed?

4.2 Graphs of Motion Constant speed means the speed stays the same. An object moving at a constant speed always creates a position vs. time graph that is a straight line.

4.2 Graphs of Motion The data shows the runner took 10 seconds to run each 50-meter segment. Because the time was the same for each segment, you know the speed was the same for each segment.

4.2 Graphs of Motion You can use position vs. time graphs to compare the motion of different objects. The steeper line on a position vs. time graph means a faster speed.

4.2 Slope The slope of a line is the ratio of the “rise” to the “run”. The steepness of a line is measured by finding its slope.

4.2 Graphs of changing motion Objects rarely move at the same speed for a long period of time. 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 Suppose we draw a rectangle on the speed vs. time graph between the x-axis and the line showing the speed. On the graph, the length is equal to the time and the height is equal to the speed. The area of the rectangle is equal to its length times its height.

Chapter Four: Motion 4.1 Position, Speed and Velocity 4.2 Graphs of Motion 4.3 Acceleration

Section 4.3 Learning Goals Define acceleration. Determine acceleration by mathematical and graphical means. Explain the role of acceleration in describing curved motion and objects in free fall.

Investigation 4B Acceleration Key Question: What is acceleration?

4.3 Acceleration Acceleration is the rate at which your speed (or velocity) changes. 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 Acceleration is easy to spot on a speed vs. time graph. Acceleration causes the line to slope up on a speed vs. time graph. What is the bike’s acceleration?

4.3 Acceleration If the hill is steeper, the acceleration is greater.

4.3 Acceleration There is zero acceleration at constant speed because the speed does not change.

4.3 Acceleration Speed and acceleration are not the same thing. You can be moving (non-zero speed) and have no acceleration (think cruise control). You can also be accelerating and not moving! A falling object begins accelerating the instant it is released.

4.3 Acceleration Acceleration describes how quickly speed changes. Acceleration is the change in speed divided by the change in time.

4.3 Speed and acceleration An acceleration of 20 km/h/s means that the speed increases by 20 km/h each second. The units for time in acceleration are often expressed as “seconds squared” and written as s2. Can you convert this rate using conversion factors?

A strong wind increases its speed to 4 m/s in 3 s. Solving Problems A sailboat moves at 1 m/s. A strong wind increases its speed to 4 m/s in 3 s. Calculate acceleration.

= 1 m/s2 Solving Problems Looking for: …acceleration of sailboat Given: …v1 = 1 m/s; v2 = 4 m/s; time = 3 s Relationships: a = v2 – v1/t Solution: a = (4 m/s – 1 m/s)/ 3 s = 1 m/s2

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

4.3 Acceleration on speed-time graphs Positive acceleration adds more speed each second. Things get faster. Speed increases over time.

4.3 Acceleration on speed-time graphs Negative acceleration subtracts some speed each second. Things get slower. People sometimes use the word deceleration to describe slowing down.

4.3 Acceleration on position-time graphs The position vs. time graph is a curve when there is acceleration. The car covers more distance each second, so the position vs. time graph gets steeper each second.

4.3 Acceleration on position-time graphs When a car is slowing down, the speed decreases so the car covers less distance each second. The position vs. time graph gets shallower with time.

4.3 Free fall 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 Falling objects increase their speed by 9.8 m/s every second, or 9.8 m/s2 The letter “g” is used for acceleration due to gravity.

4.3 Acceleration and direction Acceleration occurs whenever there is a change in speed, direction, or both.

4.3 Acceleration and direction A car driving around a curve at a constant speed is accelerating because its direction is changing.

4.3 Acceleration and direction Individual vectors can be drawn to scale to calculate the change in direction.

4.3 Curved motion A soccer ball is an example of a projectile. A projectile is an object moving under the influence of only gravity. The path of the ball makes a bowl-shaped curve called a parabola.

4.3 Curved motion Circular motion is another type of curved motion. An object in circular motion has a velocity vector that constantly changes direction.

Studying Two Part Motion Investigation 4C Studying Two Part Motion Key Question: What happens to the Energy Car as it travels down a hill and across a flat section of track?

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