### Video Transcript

An object is thrown up in the air, and it falls back down to the ground. The height, ℎ, of the object 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 four seconds?

The graph shows time in seconds on the horizontal axis and height in meters on the vertical axis. The object is being thrown up in the air and then falling back down to the ground. So, height, in this case, represents the displacement from the ground. Now, recall that speed is equal to the magnitude of the slope of a displacement–time graph. So, to calculate the speed of the ball at any particular time, we need to find the slope of the graph at that time. And we’re looking for the speed of the ball at 𝑡 equals four seconds.

So, the first thing we need to do is locate 𝑡 equals four seconds on the horizontal axis, which is down here. And then we move up from the horizontal axis to find the relevant point on our graph. Now, to find the slope of a curve at a given point, we need a tangent to the curve at that point, which is a straight line that touches the curve and has the same slope as the curve at the point where they touch. This is already shown on the graph as the dashed red line. So, to find the speed of the ball at 𝑡 equals four seconds, all we need to do is find the slope of this dashed red line.

So, recall how to find the slope of a line. The slope is the vertical difference divided by the horizontal difference between two points on the line. Now, we can use any two points on this line, but it makes sense to use ones that are easy to read on the axes. So, let’s use this one here, which is at three, 35, and this one down here, which is at five, five. So, the slope is then equal to the vertical difference between these two points, which is 35 minus five, divided by the horizontal difference, which is three minus five. 35 minus five is 30, and three minus five is equal to negative two. So, the slope of the line is negative 15.

So, we can now calculate the speed of the ball at this point, which is the magnitude of the slope. This just means the positive value of the slope. So, the positive value of minus 15 is 15. And the units are the units of the vertical axis divided by the units of the horizontal axis, which is meters per second. So, the speed of the ball at time 𝑡 equals four seconds is 15 meters per second.