Video Transcript
A ball is thrown up in the air, and it falls back down to the ground. The height ℎ of the ball above the ground over time 𝑡 is shown on the graph by the blue line. What is the speed of the ball at 𝑡 equals two seconds?
The graph shows time in seconds along the horizontal axis and the displacement as height in meters on the vertical axis. 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. And the question requires us to find the speed of the ball at a time 𝑡 equals two seconds. 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. And we can see right away that this is the point where the ball has reached its maximum height. And it is just for an instant stationary before it begins to fall back down to the ground. But let’s see how we would work this out numerically.
First, recall that speed is equal to the magnitude of the slope of a displacement–time graph. And then recall how to calculate the slope of a graph. Slope is equal to the vertical difference divided by the horizontal difference of two points on a line. 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. We can then pick any two points on that line. So let’s pick one here at zero, 20 and another one here at four, 20.
We take the second point minus the first. So the vertical difference is 20 minus 20. And the horizontal difference is four minus zero. 20 minus 20 is zero, and four minus zero is four. And zero divided by four gives us zero. 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. 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.
