Math Problem Statement
Solution
Let's go through each part of this problem step by step:
Part (a): Plotting the Graph
- You need to plot the distance (in meters) against time (in seconds).
- is plotted along the x-axis, and is plotted along the y-axis.
- From the table, you have the following values:
(s) | 0.0 | 2.5 | 5.5 | 11.0 | 16.0 | 22.0 | 26.5 | 32.0 | 39.5 | 47.5 | 56.0 |
---|---|---|---|---|---|---|---|---|---|---|---|
(m) | 0 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
- Draw a smooth curve connecting these points.
Part (b): Calculating Average Speed
Average speed can be calculated using the formula:
(i) From 0 to 10 s
- From s to s:
- ,
(ii) From 20 s to 35 s
- From s to s:
- ,
(iii) The Last 10 s (from 40 s to 50 s)
- From s to s:
- ,
Suggesting Why Average Speeds Are Different
The average speeds are different because the swimmer's velocity is not constant throughout the race. Early on, the swimmer might be accelerating, whereas later in the race, they might slow down or maintain a steady speed.
Part (c): Tangent at s
To deduce the swimmer’s instantaneous speed at s:
- Draw a tangent line to the curve at s.
- Find the gradient of this tangent line.
The gradient of the tangent will give the instantaneous speed. The steeper the line, the greater the speed at that point.
Would you like me to further explain any of these parts, or help you with graph plotting? Here are some related questions:
- What is the difference between average speed and instantaneous speed?
- How does drawing a tangent line help in finding the instantaneous speed?
- Why does the average speed change at different time intervals?
- How can acceleration be deduced from a distance-time graph?
- What is the significance of curvature in a distance-time graph?
Tip: When drawing a tangent, ensure it touches the curve at only one point without cutting across it to accurately represent the instantaneous rate of change.
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Math Problem Analysis
Mathematical Concepts
Graphing
Average Speed
Instantaneous Speed
Tangent
Rates of Change
Formulas
Average speed = Δs / Δt
Instantaneous speed is the slope of the tangent at a given point
Theorems
Rate of change on distance-time graphs
Suitable Grade Level
Grades 9-12
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