1. Some critics may sneer calculus is useless. Oh yeah? Well, we can solve this problem:
Question: The velociraptor spots you away and attacks, accelerating at from a stand start, up to its top speed of . When it spots you, you begin to flee, quickly reaching your top speed of . How far can you get before you’re caught and devoured?
2. Set up. Lets pick coordinates so you are standing at the origin on the axis. The velociraptor is at , the axis is measured in meters.
So let describe your position, and let describe the raptor’s position at time .
You run at 6 meters/second, and that can be described by
(it’s negative because you are running away from the raptor).
However, the velociraptor is accelerating at a constant rate. So
We can assume that initial velocity for the raptor is zero and again .
The problem becomes: when will ?
3. Execute. We can solve for your position immediately since (1) gives us (integrating):
Now we did something a little tricky here! We used a dummy variable in the integral.
Why didn’t we use ? Because it’s the boundary of the interval. It’d be like saying instead of .
The raptor is trickier, but lets break it up into parts. We know its acceleration. We want to find its velocity. Let be the raptor’s velocity, and . Then
This is partially true. Really, since it has a maximum speed of 25 meters/second, we should write
Observe that the velocity reaches its maximum at seconds.
How far will the raptor go? Again, we just use the fundamental theorem of calculus writing
We see that
During this time you would be at .
We now know that you’ll be devoured in under 6.25 seconds!
4. Solving. So since , we will use
Setting this equal to Equation (3) we have
This is a quadratic equation, and rearranging terms to make it prettier gives us
Using the quadratic formula, we can tell exactly when the raptor will devour you! We have two solutions
which doesn’t make sense (since the raptor spots you at time and everything takes place for ). The other solution is
which is really close to when the raptor will achieve maximum velocity!
How far did you get? Well, we see that
In other words, you got 37 meters away! (About 122 feet) Bravo!