Calculating the Area of a Triangle - KaiserScience



Name: __________________________Determining the Area on a v-t Graph plot of velocity-time gives us the acceleration of an object (the slope). Here we will learn how a plot of V-T can be used to determine the displacement of an object. For V-T graphs, the area under the curve represents the displacement. The diagram below shows three different V-T graphs; the shaded regions represent the displacement during the stated time interval.?The shaded area shows us displacement during the time t = 0 seconds to 6 seconds. This area takes on the shape of a?rectangle. It?can be calculated using the appropriate equation.??Shaded area is representative of the displacement during t = 0 to 4 seconds. This area takes on the shape of a?triangle?can be calculated using the appropriate equation.??The shaded area is representative of the displacement during from 2 to 5 seconds. This area takes on the shape of a?trapezoid?can be calculated using the appropriate equation.???How do we find the area under the curve? Geometry equations!RectangleTriangleTrapezoidArea = b ? hArea = ? ? b ? hArea = ? ? b ? (h1 + h2)?Calculating the Area of a RectangleThe rectangle on the V-T graph has a base of 6 s and a height of 30 m/s. Since the area of a rectangle is found by A = b x h, the area is 180 m (6 s x 30 m/s). So the object was displaced 180 meters, during the first 6 seconds of motion.Area = b * hArea = (6 s) * (30 m/s)Area = 180 mCalculating the Area of a TriangleThe triangle on the V-T graph has a base of 4 seconds and a height of 40 m/s. Since the area of triangle is found by A = ? * b * h, the area is ? * (4 s) * (40 m/s) = 80 m. So the object was displaced 80 meters, during the four seconds of motion.Area = ? * b * hArea = ? * (4 s) * (40 m/s)Area = 80 mCalculating the Area of a TrapezoidThe trapezoid on the V-T graph has a base of 2 seconds. Heights of 10 m/s (on the left side) and 30 m/s (on the right side). Area of trapezoid is found with A = ? * (b) * (h1 + h2)Area = 40 m [? * (2 s) * (10 m/s + 30 m/s)]. So the object was displaced 40 meters, during the time interval from 1 second to 3 seconds.Area = ? * b * (h1 + h2)Area = ? * (2 s) * (10 m/s + 30 m/s)Area = 40 mAlternative Method for TrapezoidsBreak the trapezoid into a triangle and a rectangle. The areas of the triangle and rectangle can be computed individually, and added together.Triangle: Area = ? * (2 s) * (20 m/s) = 20 mRectangle: Area = (2 s) * (10 m/s) = 20 mTotal Area = 20 m + 20 m = 40 m ................
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