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Example. Andrew has 24 square meters of ready-made lawn and plans to lay it in his garden. His wife, Jo, suggests that a good shape is a rectangle that is twice as long as it is wide. What width should Andrew make the lawn?
Solution. This is a real-life problem, but it can be solved using abstract symbols in the following way: Let the width of the lawn be x meters; its length will be 2x, because its length is twice its width (see the figure): The calculation goes as follows:
Area of rectangle = length x width
24 = 2x x x
This is an abstract equation Multiplying the terms on the right-hand
24 = 2x2
12 = x2
Dividing both sides of the equation by 2
•\/T2 = x Taking the square root of both sides of the equation
x = 3.464 Using a calculator for the square root
Now relate the abstract symbol x back to the real-life situation. The width of the lawn is 3.46 meters (three significant figures).
References: Accuracy, Königsberg Bridge Problem, Square Root.
If a car is increasing its velocity, for example, upon changing lanes on a freeway, its acceleration is the rate at which its velocity is changing with respect to time. The SI unit of acceleration is meters per second per second, which is abbreviated m s-2 or m/s2. Another unit of acceleration is centimeters per second per second, which is abbreviated cm s-2 or cm/s2.
As a body falls to earth it has an acceleration due to gravity that is approximately 10 ms-2, and this acceleration is a fixed value at different places on the earth’s surface. This means that when a stone is falling through the air, its velocity increases by 10 m s-1 for every second it is falling. In this example we ignore the air resistance. Suppose the velocity of the stone is measured every second and recorded in a table as follows:
Time t in s 0 1 2 3
Velocity v in m s-1 0 10 20 30
The values from the table are written as ordered pairs and then plotted to draw a velocity-time graph (see figure a). In this case the ordered pairs are (0, 0), (1, 10), (2, 20), and (3, 30).
0 12 3 t
The slope of this graph =------------------------- Formula for slope
30 m/s 3 s
= 10 m s-2
The acceleration of the stone is 10 m s 2.
The slope of a velocity-time graph gives the acceleration. If the stone is thrown upward, the velocity is decreasing, and the negative acceleration is called retardation. When the acceleration is not a fixed quantity, but varies, the velocity-time graph is a curve. This is the situation, for example, for a motorcyclist who accelerates from a standing position.
Example. A motorcyclist is accelerating from a standing position, and the velocities in m s-1 are recorded every 10 seconds. This information is shown in the table. The velocity of the cyclist can be expressed by the formula v = 0.03 t2. Find the acceleration of the cyclist when the time is 20 seconds.
Time in seconds 0 10 20 30
Velocity in m s_1 0 3 12 27
Solution. Using the data in the table, we draw the velocity-time graph (see figure b). The acceleration of the motorcyclist is changing, which is indicated by the curved graph. To find the acceleration at t = 20 seconds, we find the slope of the curve at the point on the graph where t is 20 seconds, and this slope gives the acceleration at that instant. The slope of the curve is the slope of the tangent to the curve at that point.
Using a ruler, we draw a tangent as accurately as possible at the point on the curve where t = 20 seconds. The tangent should just touch the curve at this point. The length of the tangent is not critical, but should not be too short, otherwise accuracy will be lost. The slope of this tangent will give the acceleration of the cyclist at 20 seconds.
Complete the right-angled triangle, and estimate the rise and the run, but take care with the units:
Slope of tangent =--------
22 m/s 20 seconds = 1.1ms“2
The acceleration of the cyclist at the time when t = 20 seconds is estimated to be 1.1m s-2.
References: Ordered Pairs, Retardation, SI Units, Slope, Tangent, Velocity, Velocity-Time Graph.
When the dimensions of a quantity are measured, such as the height of a building, the resulting measurement cannot be exact, because of the limitations of the measuring instrument. The measurement is approximate, and is given to a degree of accuracy in the form of decimal places (dp), significant figures (sf), or the nearest whole unit. Whenever measurements are made or calculations done using measured quantities, the degree of accuracy should always be stated with the answer. This process of giving an answer in approximate form is called rounding.
One day David was listening to the radio when he heard the news of an earthquake. He heard that its center was 50 kilometers below the surface of the earth. Later, he was discussing the earthquake at home; Jane had heard on the radio that the center was 48 kilometers below ground and William said it was 47.4 kilometers underground. All three measurements are correct, but differ because were rounded to different degrees of accuracy by the three different radio stations.