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Real-life graphs. We often use graphs to illustrate real-life situations. Instead of plotting y -values against x -values, we plot one physical quantity against another physical quantity. The resulting graph shows the rate that one quantity changes with another.
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Real-life graphs We often use graphs to illustrate real-life situations. Instead of plotting y-values against x-values, we plot one physical quantity against another physical quantity. The resulting graph shows the rate that one quantity changes with another. Can you think of any graphs that you have seen that are used to represent real-life situations? What quantities did these graphs use on their x-axis and the y-axis? Why?
Pounds and dollars This graph shows the exchange rate from British pounds to American dollars. It is a straight line graph that passes through the origin. The equation of the line would be of the form: y = mx. In this graph, what does the value of m represent? Using the graph, can you calculate how many dollars you would get if you had £150 to exchange?
Investing in the future investment value time This graph show the value of an investment as it gains interest cumulatively over time. The graph increases by increasing amounts. Each time interest is added, it is calculated on an ever greater amount. This makes a small difference at first, but as time goes on it makes a much greater difference. This is an example of an exponential increase.
A growing baby This graph shows the mass of a newborn baby over the first month from birth. What was the mass of the baby when it was first born? What is the baby’s mass at the end of the first month? Use the information given to describe the graph in detail.
Distance – time graphs One Sunday afternoon, John takes his car to visit a friend. • John drives at a constant speed for 30 minutes until he reaches his friend’s house 20 miles away. • He stays at his friend’s house for 45 minutes. • He drives home at a constant speed and arrives home 45 minutes later. Can you draw a graph to represent John’s journey? What quantity will you put on the x-axis? What quantity will you put on the y-axis? Why?
Finding speed change in distance distance change in time time change in distance change in time From our speed, distance, time triangle, we know that speed is calculated by dividing distance by time. How do we calculate speed from a distance – time graph? gradient = = speed What does a zerogradient mean for the object’s speed?
Distance – time graphs change in speed time When a distance – time graph is linear, the objects involved are moving at a constant speed. Most real-life objects do not behave like this. They are far more likely to speed up and slow down during a journey. Increase in speed over time is called acceleration. acceleration = It is measured in metres per second per second or m/s2. A decrease in speed over time is called deceleration.
Acceleration and deceleration distance time distance time Distance – time graphs that show acceleration or deceleration have a curved appearance to them. This distance – time graph shows an object accelerating from rest before continuing at a constant speed. This distance-time graph shows an object decelerating from a constant speed before coming to rest.
Speed – time graphs Travel graphs can be used to show change in speed over time. • This graph shows a car accelerating steadily from rest to a speed of 20 m/s. • It then continues at a constant speed for 15 seconds. • The brakes are then applied and it decelerates steadily to a stop. How would you calculate the acceleration and deceleration of the car?
Acceleration from speed – time graphs speed change in speed change in time time change in speed change in time Acceleration is calculated by dividing speed by time. In a speed – time graph, this is the gradient of the graph. How do we calculate acceleration from a speed – time graph? gradient = = acceleration A negative gradient means that the object is decelerating.
Distance from speed – time graphs The following speed – time graph shows a car driving at a constant speed of 20 m/s for 2 minutes. What is the area under the graph? Area under graph = 20 × 120 = 240 What does this amount correspond to?
The area under a speed – time graph This speed – time graph shows a car accelerating, travelling at a constant speed and then decelerating to a stop. What distance has the car travelled? Show your working.
Temp (°C) Temp (°F) What’s the temperature? Frank has been asked to draw a graph that illustrates the temperature relationship between °F and °C. Frank records the following information from his research. 0 20 40 60 80 100 212 32 68 104 140 176 What is the temperature in °F when it is 70°C? What is the gradient of the graph when °F is plotted on the vertical axis? Can you find a way to express the relationship between °F and °C?
A good deal? Theo is looking for a new mobile phone and has seen the model he wants advertised on two different tariffs. 12 month contract: £9.99 a month FREE handset and texts! Calls only 10p per minute! PAYG £40 for handset FREE texts Calls only 5p per minute! Which tariff is better value if Theo makes 200 minutes of calls in the first month? Show your working. At what stage in the first month does the monthly contract cost more than the PAYG phone?
