An object is at rest at #(8 ,2 ,9 )# and constantly accelerates at a rate of #3 m/s# as it moves to point B. If point B is at #(6 ,7 ,5 )#, how long will it take for the object to reach point B? Assume that all coordinates are in meters.
The time is
Apply the equation of motion
The initial velocity is
The acceleration is
Find the shortest distance between point A and B, then using calculus, use the definition of acceleration and velocity to solve for the displacement in terms of time from the given acceleration, then solve for time using the previously calculated shortest distance, to get
Rephrasing the question, we have an object at point A,
But first, let's talk about the initial displacement. We can see that, from point A to B, the object has moved:
Now, we just need the time, but we may have to play around with acceleration and velocity for that. We don't yet know the velocity, and we can take the integral of acceleration to get that...
... But wait! The
It may feel weird that I already have it set up for all three variables even though the Pythagorean theorem is originally for two variables. However, it really is just a result of attempting to apply the Pythagorean theorem twice:
Anyways, we could now substitute in for
Ah, so the acceleration was
So, using the definition of acceleration:
We know that
Ah, the constant! Well, it's supposed to be whatever the starting velocity happens to be. Since we started with no motion at all - no displacement, no velocity, no acceleration - then this
Then, what is the definition of velocity?
Now, plug in for
Again, the constant is just zero:
However, we do know what the displacement is! Substitute
Interesting, we can now take the square root:
That's a werid amount of seconds, and it's certainly an irrational number... we could approximate it using a calculator:
There we go! That's what our calculations tell us about the time it takes for an object with the given acceleration and initial state to travel through the two given points.