Before we get started, let't think about one thing - Will the velocity of the swimmer be a constant value? The answer to this is ABSOLUTELY NOT!
The velocity of the swimmer is not a constant. When the swimmer push off the deck, his or her velocity will be zero and quickly increase to its maximum value and decrease to zero because the swimmer has reached the other side of the swimming pool. When the swimmer turns around to finish the other 25 meters, because the swimmer is going to the opposite direction, his velocity will be a negative value which will soon reach its minimum value. When the swimmer finish the race, the velocity of the swimmer will decrease to zero again.
The derivative of distance is the swimmer's velocity
The derivative of distance is the swimmer's velocity
How to find the equation of velocity over time?
To find the equation of velocity over time is not complicated. In physics, to find the velocity over time to find the derivative of the distance over time. (Remember V=d/T)
Having said that, I have found the equation of velocity over time using the First Principle method (Calculation is at the end of the page.)
Now we found the equation, what does this mean in terms of swimming?
The amplitude of this equation is 2 and since there is no vertical translations, this means from the equation of the mid-line (y=0) to the maximum point and the minimum point is 2. This also tells up that the maximum velocity of the swimmer is 2m/s.
The Period of this equation is 2π/38, means that the swimmer takes 38 seconds to complete one complete cycle of swimming and the velocity of the swimmer goes back to the original velocity when the swimmer started (0m/s).
Having said that, I have found the equation of velocity over time using the First Principle method (Calculation is at the end of the page.)
Now we found the equation, what does this mean in terms of swimming?
The amplitude of this equation is 2 and since there is no vertical translations, this means from the equation of the mid-line (y=0) to the maximum point and the minimum point is 2. This also tells up that the maximum velocity of the swimmer is 2m/s.
The Period of this equation is 2π/38, means that the swimmer takes 38 seconds to complete one complete cycle of swimming and the velocity of the swimmer goes back to the original velocity when the swimmer started (0m/s).
Transformation, Min & Max, and the Curve of the shape.
The amplitude of this function is two, which means that the sinusoidal function is vertically stretched by a factor of 2. However there is no vertical translations, so this also means that the maximum velocity that the swimmer can have is 2. The period of this function is 2pi/38, which means that the graph is horizontally stretched by a factor of 38/2pi. This means that the speed of the swimmer will come to its original speed (0,0).
The shape of the curve between the start of the graph, it is increasing until the swimmer reaches the maximum velocity. He slows down to 0, which the curve of the graph is decreasing. The graph kept on decreasing after the swimmer reaches the other end of the pool because he is going in a opposite direction which means negative velocity. After the minimum point, the curve of the graph started to increase, by this, the velocity of the swimmer is actually decreasing. Until it deceases to zero velocity.
The shape of the curve between the start of the graph, it is increasing until the swimmer reaches the maximum velocity. He slows down to 0, which the curve of the graph is decreasing. The graph kept on decreasing after the swimmer reaches the other end of the pool because he is going in a opposite direction which means negative velocity. After the minimum point, the curve of the graph started to increase, by this, the velocity of the swimmer is actually decreasing. Until it deceases to zero velocity.
Let's take a look at the relationship between two equations
(optional to read but this helps explaining why the distance over time is a sinusoidal function!)
The BLUE graph represent the equation of distance over time, and the RED graph represent the equation of velocity over time. When we put these two graphs together, we see some interesting points. The GREEN points marked on the graph has special meanings.
Lets start from the first GREEN point. The coordination of both f(x) and V(x) are (0,0). This simply means that the swimmer starts to swim at a velocity of 0, time of 0, distance of 0. This is a reasonable aspect in terms of swimming.
Lets start from the first GREEN point. The coordination of both f(x) and V(x) are (0,0). This simply means that the swimmer starts to swim at a velocity of 0, time of 0, distance of 0. This is a reasonable aspect in terms of swimming.
The second GREEN point on the BLUE graph is (9.5, 12.5) and the GREEN point on the RED graph is (9.5, 2). This means that the swimmer is half way to the end of the swimming pool (12.5 meters) and he has a velocity of 2 m/s which is his maximum velocity. This is an accurate result because the time of both points is the same-9.5 s. From the first point to the second point, the velocity of the swimmer has been increasing.
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The third GREEN point on the BLUE graph is (19, 25) and the GREEN point on the RED graph is (19,0). This means that the swimmer reach the end of the pool and his velocity decreased to 0 m/s. This is an accurate result because the time for both points is 19 s. From the second point to the third point, the velocity of the swimmer has been decreasing because he is almost reached the end of the pool.
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The fourth GREEN point on the BLUE graph is (28.5,12.5) and the GREEN point on the RED graph is (28.5, -2). This means that the swimmer reached the half way point of the other end of the pool (where he started). His velocity is -2m/s because he is swimming in the opposite direction as he started. This is an accurate result because the time for both points is 28.5 s. From the third point to the fourth point, the velocity of the swimmer has been decreasing to -2m/s (which in real life is 2m/s) because he is swimming in an opposite direction.
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The last GREEN point. The coordination of both f(x) and V(x) are (0,0). This simply means that the swimmer finished 50 meters and came back to where he originally started. At a velocity of 0, time of 0, distance of 0. This is a reasonable aspect in terms of swimming.
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