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. The red line is a tangent to the blue line at 𝑡 equals one second. What is the speed of the ball at 𝑡 equals one second?
Now, recall that instantaneous speed is the magnitude of the slope of a displacement–time graph. Now, if we refer to our graph, we have time 𝑡 on the horizontal axis and height ℎ on the vertical axis. In this case, the height refers to the height above the ground of the ball as it is thrown in the air and falls back down. And so this is our displacement.
So what we’re looking for is the magnitude of the slope of this graph at a time of 𝑡 equals one second. So let’s start by finding 𝑡 equals one second on our horizontal axis. And we work upwards from the horizontal axis to find our blue line at this point. To find the slope of a curve at a specific point, we need a tangent at that point. And the question helpfully provides one, the red line, which is a tangent to the blue line at 𝑡 equals one second. That means it touches the blue line at 𝑡 equals one second and has the same slope as the blue line at that point.
Therefore, what we need to find is the magnitude of the slope of the red line. Now, recall that the slope is the vertical difference divided by the horizontal difference between two points on a line. A straight line has a constant slope. So we could use any two points to work this out. But it makes sense to use points that are easy to read on the axes. Let’s try these two points. These are zero, five and two, 15. So the vertical difference is 15 minus five, and the horizontal difference is two minus zero. 15 minus five is 10, and two minus zero is two. So the slope of this line is 10 divided by two, which is five.
And the units are the units on the vertical axis, which are meters, divided by the units on the horizontal axis, which are seconds. And the magnitude is just a positive value of this number, which is five. So the speed of the ball at 𝑡 equals one second is five meters per second.
