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A ball is thrown up in the air, and it falls back down to the ground.
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The height ℎ of the ball above the ground over time 𝑡 is shown on the graph by the blue line.
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What is the speed of the ball at 𝑡 equals two seconds?
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The graph shows time in seconds along the horizontal axis and the displacement as height in meters on the vertical axis.
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The blue line shows the ball starting from the ground, rising in the air as its height increases, coming to a stop here, and then falling back down to the ground.
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And the question requires us to find the speed of the ball at a time 𝑡 equals two seconds.
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So let’s start by finding 𝑡 equals two seconds on the horizontal axis and then working up from the axis to find the ball at this point.
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And we can see right away that this is the point where the ball has reached its maximum height.
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And it is just for an instant stationary before it begins to fall back down to the ground.
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But let’s see how we would work this out numerically.
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First, recall that speed is equal to the magnitude of the slope of a displacement–time graph.
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And then recall how to calculate the slope of a graph.
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Slope is equal to the vertical difference divided by the horizontal difference of two points on a line.
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So we could draw a tangent to this line at a point 𝑡 equals two seconds, which is a straight line that touches a curve and has the same slope as the curve at the point where they touch.
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We can then pick any two points on that line.
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So let’s pick one here at zero, 20 and another one here at four, 20.
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We take the second point minus the first.
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So the vertical difference is 20 minus 20.
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And the horizontal difference is four minus zero.
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20 minus 20 is zero, and four minus zero is four.
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And zero divided by four gives us zero.
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For the units, we take the units of the vertical axis, so that’s meters, and then divide by the units of the horizontal axis, which are seconds.
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So the speed of the ball at time 𝑡 equals two seconds is zero meters per second, or in other words the ball is stationary.