# Finding Speed at a Single Point

I am given this information:

A question asks me to estimate the speed at point P to the nearest integer.

I understand that this has something to do with the tangent line at P, but I'm not sure how to find it. Any help?

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You can try extrapolating the part between $Q_4$ and $P$ to get an estimate. Between $Q_4$ and $P$ the increase in time is $20 - 19.3 = 0.7$ seconds, while the increase in distance is $650 - 614 = 36$ meters. So in that interval, the object travels at $36$ meters per $0.7$ seconds, or $\frac{36}{0.7} \approx 51.43$ meters per second. The speed at $P$ may be a bit more looking at the shape of the function, but $51$ or $52$ meters per second should be a pretty good estimate.

Edit: We could try to fit a quadratic with the given data, but there is no exact match. Plugging in the data in Mathematica,

Fit[{{8, 165}, {13, 333}, {18.2, 559}, {19.3, 614}, {20, 650}}, {1, x, x^2}, x]


it spits out

0.976595 x^2 + 13.0624 x - 1.96826


Evaluating it at those five points, we get small errors:

{165.033, 332.887, 559.255, 613.908, 649.918}


We can then simply take the derivative w.r.t. $x$ to get the slope (speed) at any point $x$. The derivative is $1.95319 x + 13.0624$, so then plugging in $x = 20$ also gives us that the speed at the point $P$ is roughly $52.1262$. So again, $52$ meters per second seems like the best answer.

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@Mike: Unless you have an exact formula for the curve, you can only approximate it. – TMM Sep 24 '11 at 13:41
For the record, you can get the same value in Wolfram|Alpha by inputting "D[Fit[{{8, 165}, {13, 333}, {18.2, 559}, {19.3, 614}, {20, 650}}, {1, x, x^2}, x],x] /. x -> 20" – TMM Sep 24 '11 at 14:02

The shape of the curve and the data show that we have something roughly quadratic. Assume the curve has equation $Ax^2+Bx+C$ and find the values of A, B and C from the data. Then a bit of calculus will give you the gradient of the curve at P and hence the speed.

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[Note: I'm not going to give all of the details, because this is a homework question]

Find the average velocity between Q1 and Q2, and assume that it represents the velocity half-way between them. Then repeat for each of the following pairs of points.

Then look at the trend of velocity vs position. You will find that it follows a simple relationship. You can then follow that trend up to point P, to estimate the velocity at point P.

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