Athlete runs twice round the circular athletics track. The track has a total length of 320 m and a diameter of 100 m. Make a scale drawing of the track. Use a scale of 1 cm represents 10 m. Taking the start as point 1, find both the athlete's distance and displacement at points 1, 2, 3 and 4, relative to the start, for both laps that he completes. Write down the athlete's exact displacement when he passes point 3 on the track.
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To make a scale drawing of the track, we need to use the given scale of 1 cm represents 10 m. Since the total length of the track is 320 m, we can calculate the length of the drawing by dividing 320 by 10, which gives us 32 cm. Therefore, the scale drawing of the track will have a length of 32 cm.
Next, we can draw a circle to represent the track. Since the diameter of the track is given as 100 m, the radius of the circle will be half of that, which is 50 m. In the scale drawing, 1 cm represents 10 m, so the radius of the track in the scale drawing will be 5 cm. Draw a circle with a radius of 5 cm to represent the track.
Now, let's label the points on the track for easier reference. Label the starting point as 1, and continue labeling the points clockwise as 2, 3, and 4.
To find the athlete's distance and displacement at each point for both laps, we need to calculate the actual distance and displacement using the given scale.
The distance is simply the length of the path covered by the athlete. Since the athlete runs twice round the track, the distance at each labeled point will be twice the length of the track. In this case, the distance at each point will be 2 * 320 m = 640 m.
The displacement, on the other hand, is the change in the position of the athlete relative to the starting point. It is a direct line from the start point to the current point. We can use the Pythagorean theorem to calculate the displacement at each point.
For example, at point 1, the displacement will be the radius of the track since the athlete is still at the starting point. The displacement at point 1 will be 5 cm in the scale drawing, which represents 10 m in reality.
Similarly, at point 2, the displacement will be the hypotenuse of a right triangle with one leg being the radius (5 cm) and the other leg being the diameter of the track (10 cm in the scale drawing). We can use the Pythagorean theorem to calculate the displacement at point 2:
displacement^2 = radius^2 + diameter^2
displacement^2 = 5^2 + 10^2
displacement^2 = 25 + 100
displacement^2 = 125
displacement ≈ √125 ≈ 11.2 cm in the scale drawing, which represents 22.4 m in reality.
Similarly, we can calculate the displacement at points 3 and 4 using the same method.
To find the athlete's exact displacement when passing point 3 on the track, we need to calculate the displacement from the start point to point 3 directly. Again, we can use the Pythagorean theorem to calculate the displacement:
displacement^2 = radius^2 + (2 * radius)^2
displacement^2 = 5^2 + (2 * 5)^2
displacement^2 = 25 + 100
displacement^2 = 125
displacement ≈ √125 ≈ 11.2 cm in the scale drawing, which represents 22.4 m in reality. Therefore, the athlete's exact displacement when passing point 3 on the track is 22.4 m.
In summary, the athlete's distance at each point for both laps is 640 m, and the athlete's displacement at points 1, 2, 3, and 4 for both laps is approximately 10 m, 22.4 m, 22.4 m, and 22.4 m respectively.