Peter's Physics Pages

Bridging Course - Lecture 1 (position and speed)

In this lecture the following are introduced:
• Physics as the nature of things
• Knowledge and Belief (Zeno's paradox)
• The Physics Bridging Coure goal
• Average and Instantaneous Speed
• Position/time graph to speed/time graph
• speed/time graph to position/time graph

Physics as the nature of things.

As you experience the world around you, your mind automatically does a kind of Physics; it works out the nature of things, like walking, climbing and throwing, then it helps you to do them. However to transform these innate abilities into simple rational statements of universal application, requires a body of knowledge with the right organising principles to interpret and structure your observations and concepts.

Knowledge and Belief

The study of motion got off to a false start with Zeno's paradox of Achilles and the tortoise.

In an imaginary race between a fast runner (Achilles) and a slow runner (a tortoise) the tortoise is given a head start. After the start Achilles runs at constant pace, and while he does that the tortoise moves at a constant but slower speed. When Achilles gets to where the tortoise was initially T0, the tortoise has moved ahead to T1. Achilles doesn't pause where the tortoise was ( T0), and runs to where the tortoise now is ( T1), but during that time the tortoise moves ahead (to T2), not as far as Achilles did but has still moved ahead. Achilles continues to where the tortoise now is ( T2), but during that time the tortoise has moved on and remains ahead (at T3).

Achilles doesn't pause where the tortoise was, and runs to where the tortoise now is, but during that time the tortoise moves ahead, not as far as Achilles did but the tortoise is still ahead.

This argument can repeat indefinitely with an infinite number of time intervals, so reasoning this way, there is no time when Achilles will catch up to the Tortoise. This we know is wrong from experience. Clearly there is a fault in the reasoning. The fault lies in a natural belief that the sum of an infinite number of time intervals will be an infinite time. Although it may not be obvious, the sum of an infinite number of time intervals can be a finite time. The mathematical explanation of this is at the end of Introductory Physics lecture 15.

There were other false starts in Physics, such as the belief that the Sun, stars and planets revolved around a stationary Earth.

The lesson from these is that you need knowledge, and not just belief, if you want to understand the real nature of things. Knowledge (Latin: scientia) comes from experience and training. Physics itself limits experience to the directly or indirectly observable.

The Physics Bridging Course Goal

The Physics Bridging Course is for those who feel that their preparation for First Year University may not be adequate and serves as an introduction to the subject. In this course you will start to learn the language, style and techniques of Physics by looking at:

 Kinematics How to describe motion Dynamics How motion is caused by force, and The effect of force through time Energy The effect of force through space Electricity Electrical force and energy

Kinematics

Kinematics (or "cinematics" with a soft "c") means "moving pictures", so kinematics is how the various aspects of motion are described in time. To do this, mathematical points have to be uniquely specified both in space and time Rene Descartes (1596 to 1650) argued that with a single reference point and a reference direction every point in a plane (or in space) can be uniquely identified. The Cartesian co-ordinate system has a reference point (or "origin"), a reference direction "x", and one (or two) perpendicular axes, "y" (for a plane) and "z" (for three dimensional space). A point in space is specified by co-ordinates along each axis, as shown on the left. He continued with this approach and created what is known as Coordinate Geometry.

 To introduce time, the horizontal axis is taken to give sequential moments in time and the vertical axis represents a direction in space. Two space directions ("x" and "y") can be reasonably represented in a "three dimensional" diagram, but three space axes takes a little imagination! In the diagram on the right, the co-ordinates (t,x) or (x,t) at a point on the line gives the straight-line distance of an object from its origin along the "x" axis, at the moment in time "t". Definition of average speed

An object's average speed between two times is defined by how quickly the position changes during that time interval, i.e.  This will have the S.I. unit of m.s-1, however in traffic regulations, km/h is used. (Note that non-S.I. units have a "forward slash" and not an index).

Example 1
A car is driven from Sydney to Byron Bay, a distance of 910 km in 13 hours. Find the average speed for the trip. On the trip, the car travelled at 70 km/h at least on two occasions, but it would have varied a lot during those 13 hours. To deal with this, another concept is needed, the speed at an instant of time i.e. the instantaneous speed.

Definition of instantaneous speed

The instantaneous speed is the slope of the tangent to a position vs time graph at a particular time. By using the slope of the tangent at a point, we avoid the difficulty of producing a time interval at a single point in time. Notice the symbolism, "v" is instantaneous speed and the slope of the tangent is represented by "dx/dt". In reading such equations it never pays to focus on the symbols. That is, don't think "vee equals dee ex over dee tee" always read it in your mind as the instantaneous speed is given by the slope of the tangent on the position vs time graph. For example, putting scales on the axes of the diagram (as shown on the left) and selecting a point (2s, 1m) the instantaneous speed can be calculated by drawing a tangent. In this case the distance would change by 0.9m over 1.2s giving an instanateous speed of 0.75 ms-1.

Finding the speed vs time graph from the position vs time graph.

By working out the slopes at every point on a position vs time graph, the corresponding instantaneous speed vs time graph can be constructed. In interpreting these graphs, we can see that, on the position vs time graph, the steeper the slope the higher the speed.

Example 2
Comment on the instantaneous speeds at the four points labelled. At t1 the speed is slow. At t2 the speed is fast. At t3 the speed is zero. At t4 there is reverse speed.

Finding the position vs time graph from the speed vs time graph.

If you start from home and drive at a constant speed of 60 km/h for 2 hours, you will be 120 km from home. The speed vs time graph is shown on the left. The effect of speed through time is to change the position. By re-arranging the definition of average speed,distance travelled = (average speed) × (time involved). This product represents an area. A position vs time graph can be drawn from the areas under a speed vs time graph. Blue areas become red vertical lines. Each of the small blue areas is 60 × 0.5 which gives a position change of 30km. Note that this is a change in position and not an absolute position. The total blue area gives a position change of 60 × 2 which is 120km, the height of the line at 2h.

Position vs Speed graphs summary The position vs time graph shows the position at any time as the curvey red line. By measuring slopes at each point (e.g. the blue sloping line) the speed can be measured at each time, (shown as the vertical blue line in the lower graph). By measuring areas on the speed vs time graph (e.g. the red area) the change in position can be shown as the vertical red arrow in the upper graph.

The two techniques of finding slopes (how quickly something changes in time or space) and
finding areas (the effect of something through time or space), will dominate all further understanding in Physics.

Example 3
An object changes its position as shown. Find
(a) the speed between 0s and 1s, and
(b) the instantaneous speed at 3s. (a) speed between 0s and 1s (b) instantaneous speed at 3s Example 4
Starting +2m from its origin, an object's speed is given by the following graph. Find the position of the object at 5s. Between 0s and 1s: Change in position = area of triangle between 0 and 1 = ½ × 1 × 1 = 0.5m Between 1s and 3s: Change in position = area of rectangle between 1 and 3 = 2 × 1 = 2m Between 3s and 5s: Change in position = area of rectangle between 3 and 5 = 2 × 2 = 4m Total distance travelled is 6.5 m. However the object starts at +2m so its position is +8.5 m from its origin.

Summarising:

Physics is the nature of things.
Physics requires a body of knowledge with the right organising principles to interpret and structure your observations and concepts.

The two techniques of finding slopes (how quickly something changes in time or space) and finding areas (the effect of something through time or space), will dominate all further understanding in Physics.  The instantaneous speed is the slope of the tangent to a position vs time graph at a particular time. An speed vs time graph can be constructed from the slopes of a position vs time graph. A position vs time graph can be drawn from the areas under a speed vs time graph. email Write me a note if you found this useful

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