Chapter 4 Linear Motion

4.1 Motion is Relative

When we describe the motion of one object with respect to another, we say the object is moving relative to the other object.

  • Ex. A book that is at rest, relative to the table it lies on, is moving at about 30 km/s relative to the sun.

An object is moving if its position relative to a fixed point is changing.

4.2 Speed

Galileo is credited as being the first to measure speed by considering the distance covered and the time it takes. Speed is how fast an object is moving.

You can calculate the speed of an object by dividing the distance covered by time.

Speed = distance/time

Instantaneous Speed

The speed at any instant is called instantaneous speed.

  • Ex. A car does not always move the same speed. A car may travel down the street at 50km/h, slow to 0 km/h at a red light, and speed up to only 30km/h because of traffic.

Average Speed

The average speed is the total distance covered divided by the time.

Average speed = total distance covered/time interval

If we know the average speed and travel time, the distance is easy to find.

total distance covered = average speed x travel time

4.3 Velocity

Velocity is speed in a given direction. We can say a car travels at 60km/h and it would describe speed but if we say a car travels at 60km/h north then it would be velocity.

Speed is a description of how fast an object moves; velocity is how fast and in what direction it moves.

A quantity such as velocity that specifies direction as well as magnitude is called a vector quantity because a vector quantity needs both magnitude and direction for completion.

Constant Velocity

Constant speed means steady speed, speed that doesn’t slow down or speed up. Constant velocity means both constant speed and direction. Constant direction is a straight line—the object’s path doesn’t curve.

Changing Velocity

If either the speed or the direction (or both) is changing, then the velocity it changing. Constant speed and constant velocity are not the same, a body may move at a constant speed along a curved path but because the direction isn’t constant, neither is the velocity.

4.4 Acceleration

Acceleration is the rate at which the velocity is changing.

You can calculate the acceleration of an object by dividing the change in its velocity by time.

Acceleration = change in velocity/time

The key idea that defines acceleration is change. Having good acceleration means being able to change velocity quickly and does not necessarily refer to how fast something is moving.

The term acceleration applies to both decreasing as well as increasing in speed.

  • Ex. the brakes of a car can produce large retarding accelerations, that is, they can produce a large decrease per second in speed. This is called deceleration. We experience deceleration when the driver of a bus or car slams on the brakes and we tend to hurtle forward.

Change in Direction

Acceleration also applied to changes in direction. If you ride around a curve at a constant speed of 50km/h, you feel the effects of acceleration as your body tends to move toward the outside of the curve. You may round the curve at a constant speed but your velocity isn’t constant because your directions is changing every instant. If you’re state of motion is changing, then you are accelerating. Acceleration is defined as the rate of change in velocity, rather then speed. Acceleration, like velocity, is a vector quantity because it is directional.

Changing in Speed

When a straight line motion is considered, it is common to use speed and velocity interchangeably. When the direction is not changing, acceleration may be expressed s the rate at which speed changes.

Acceleration (along a straight line) = change in speed/time interval

4.5 Free Fall: How fast

  • Ex. an apple falls from a tree. Does it accelerate while falling? We know it starts from rest and gains speed as it falls. We know this because it would be safe to catch if it fell from a meter or two, but no if it fell from a high-flying balloon. Thus, the apple must gain more speed during the time it drops from a greater height than during the shorter time it takes to drop from a meter.

This gain in speed indicates that the apple does accelerate as it falls.

Falling Objects

Gravity causes the apple to accelerate downward once it begins to fall. In real life, air resistance affects the acceleration of a falling object.

  • Ex. Imagine there is no air resistance and that gravity is the only thing affecting a falling object. An object under the influence of gravitational force is said to be in free fall.

Free falling objects are affected only by gravity. The elapsed time is the time that passed once the beginning of any motion, in this case the fall.

The acceleration of an object in free fall is about 10 meters per second squared (10m/s2)

Rising Objects

Consider an object being thrown straight up. At the highest point, when the object is changing direction of motion from upward to downward, its instantaneous speed it zero. Then it starts downward just as if it had been dropped from rest at that height.

During the upward part of this motion, the object slows from its initial upward velocity to zero velocity. We know the object is accelerating because its velocity is changing.

The speed decreases at the same rate it increased.

4.6 Free Fall: How far

Speed and distance are not the same thing. At the end of the first second, the falling object has an instantaneous speed of 10 m/s. The initial speed of the fall is zero and takes a full second to get to 10m/s. So the average speed is halfway between zero and 10m/s—thats 5m/s. So during the first second, the object has an average speed of 5m/s and falls a distance of 5m.

Accelerating objects don’t have to be free falling objects.

  • Ex. A car accelerates when you push on the gas or break pedal.

Whenever an objects initial speed is zero and the acceleration (a) is constant and steady, the equations for velocity and distance traveled are:

V = at and d = ½at2

4.7 Graphs of Motion

LOOK AT TESTS, HOMEWORK, AND PAST ASSIGNMENTS FOR UNDERSTANDING

4.8 Air Resistance and Falling Objects

Air resistance noticeably slows the motion of things with larger surface areas like fallings feather or pieces of paper. But air resistance less noticeably affects the motion of more compact objects like stones and baseballs. In many cases the effect of air resistance is small enough to be neglected. With negligible air resistance, falling objects can be considered to be free falling.

4.9 How fast, How far, How Quickly How Fast Changes

When we wish to specify how fast something freely falls from rest after a certain elapsed time, we are talking about speed and velocity.

V = gt

When we wish to specify how far that object has fallen, we are talking about distance.

d = ½gt2

Velocity or speed (how fast) and distance (how far) are entirely different.

One of the most confusing concepts is acceleration, or “how quickly does speed or velocity change). What makes acceleration so complex is that its a rate of a rate. It is often confused with velocity, which is a rate itself (the rate at which distance is covered). Acceleration is not velocity, nor is it even a change in velocity.

Practice Problems

4.1

How can you be both at rest and also moving about 107,000km/h at the same time?

You cover 10 meters in a time of 1 second. Is you speed the same if you cover 20 meters in 2 seconds?

4.2

  1. Does a speedometer of a car read instantaneous speed or average speed?

4.3

  1. Which is a vector quantity? speed or velocity?

  2. What two controls on a car cause a change in speed? What control causes only a change in velocity?

4.4

  1. What is the acceleration of a moving car moving along a straight-line path that increases its speed from zero to 100 km/h in 10 seconds?

  2. By how much does the speed of a vehicle moving in a straight line change each second when it is accelerating at 2km/hxs? at 4km/hxs? at 10km/hxs?

  3. Why does the unit of time enter twice in the unit of acceleration?

4.5

  1. What is the meaning of free fall?

  2. For a free falling object dropped from rest, what is the instantaneous speed at the end of the fifth second of fall? The sixth second?

  3. For a free falling object dropped from rest, what is the acceleration at the end of the fifth second of fall?

4.6

  1. How far will a free falling object fall from rest in five seconds? Six seconds?

  2. How far will an object move in one second if its average speed is 5m/s?

  3. How far will a freely falling object have fallen from a position of rest when its instantaneous speed is 10m/s?

4.7

  1. What does the slope of the curve on a distance vs. time graph represent?

  2. What does the slope of the curve on a velocity vs. time graph represent?

4.8

  1. Does air resistance increase or decrease the acceleration of falling object?

4.9

  1. What is the appropriate equation for how fast an object freely falls from a position of rest? For how far that objects falls?

Plug & Chug + Think & Explain

Average speed = total distance covered/time interval

  • Calculate your average walking speed when you step 1 meter in 0.5 seconds.

  • Calculate the speed of a bowling ball that moves 8 meters in 4 seconds.

  • Calculate your average speed if you run 50 meters in 10 seconds.

Distance = average speed x time

  • Calculate the distance that Charlie runs if he maintains an average speed of 8km/h for 1 hour.

  • Calculate the distance you will travel if you maintain an average speed of 10 m/s for 40 seconds.

  • Calculate the distance you will travel if you maintain an average speed of 10km/f for ½ hours

Acceleration = change of velocity/time interval

  • Calculate the acceleration of a car that can go from rest to 100km/h in 10 seconds.

  • Calculate acceleration of a bus that goes from 10km/h to a speed of 50km/h in 10 seconds.

  • Calculate the acceleration of a ball that starts from rest and rolls down a ramp and gain a speed of 25 m/s in 5 seconds.

Instantaneous speed = acceleration x time

V = at

  • Calculate the instantaneous speed at the 10 second mark for a car that accelerates at 2 m/s2

  • Calculate the speed of skateboarder who accelerates from rest for 3 seconds down a ramp at an acceleration of 3 m/s2

Velocity acquired in free fall from rest:

V = gt

  • Calculate the instantaneous speed of an apple 8 seconds after being dropped from rest.

  • A sky diver steps from a high-flying helicopter. If there were no air resistance, how fast would she be falling at the end of a 12-second jump?

D = ½gt2

  • Calculate vertical distance an object dropped from rest would cover in 12 seconds if it fell freely without air resistance.