Modeling free fall with quadratic functions
About this video
A glove falls from a ski lift 65 meters above the ground, and 4.9t^2 gives the distance it has fallen after t seconds. The height is the starting height minus the distance fallen, h(t) = 65 - 4.9t^2. At t = 3 the height is 20.9 meters, which matches the graph.
You'll be able to
- tell the difference between distance fallen and height above the ground.
- write a height function h(t) = starting height - 4.9t^2.
- evaluate the function and check the answer against a graph.
Key words
- distance fallen
- How far an object has dropped since it started falling.
- height
- How far an object is above the ground.
- free fall
- Falling under gravity alone; here the distance fallen is 4.9t^2 meters.
Transcript
- [Instructor] We're told the expression 4.9 t squared represents the distance in meters, an object falls after t seconds. A glove falls from a ski lift that is 65 meters above the ground. The graph shows the height h in meters of the glove t seconds after it falls.
So that's this graph right over here. So it looks like at t zero we're at 65 meters. That's what they said. And it looks like we hit the ground a little bit more than three and a half seconds afterwards. Fair enough. Write a function that represents the height of the glove in meters after t seconds.
Well, we know what the distance it falls in after t seconds. They tell us right there, the expression 4.9 t squared represents the distance in meters, an object falls. So you might be tempted to say, "Okay, I'll just type in 4.9 t squared," but this would not be correct 'cause this is the distance that the object falls, not its height.
So its height would be wherever it started, minus the distance it falls. So in this example, where did it start? Well, it started at a height of 65 meters and they told us that right over here, a glove falls from a scale up that is 65 meters above the ground. So what you would want to do is take your starting height, 65 meters, and then subtract out the distance that it has fallen.
So there you here, this is a function that represents the height h of t. And I guess if I wanted to write it as a function, I'd say h of t is equal to that, to make sure that I'm not just writing expression, h of t is equal to that. And then I have complete the table by entering the height at t equals three seconds.
Round your answer to the nearest 10th of a meter. So just to think about what's going on. If I look at this graph at three seconds, it looks like we're a little bit more than 20 meters in height. And let's see if this function we created gives us that same answer.
And to help us there, let me get a calculator out. And so we're talking about at three seconds. So let's take three squared. I'm just putting three for t here. Three squared times 4.9, times 4.9 is equal to this, and we wanna subtract that from 65. So lemme just make that negative and add to that 65, and I get 20.9 meters and that validates what we saw on the graph as well.
A little bit more than 20 meters. And we're all done.
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