V(t) models the velocity of a spaceship. Find the position of the spaceship after 5 seconds if P(t) (the position function) contains the point (2,15).

Remember how position/velocity/acceleration are all related: Velocity is the first derivative of position, and acceleration is the derivative of velocity. 

So to get to position, we need to integrate velocity.

To integrate polynomials, we increase the exponent by 1, and then divide  by the new exponent.

Thus, our function so far looks like:

However, we know that it must pass through the point (2,15), so we have to plug the point in and solve for C

So our function is:

Then, to find it after 5 seconds:

To approach this problem, bear in mind that there are two states for Harold; one in which he's accelerating, and another in which his speed is held constant. Begin with calculating how far he travels from the stone while accelerating:

Since he begins at rest

And make note of his velocity at the five second mark:

Now, to find where he travels while accelerating, integrate once more:

We are told he's initially  feet from a stone:

This is his position after he travels while accelerating. Now to find how far he travels for the one second while his speed is constant, treat this position as a new initial condition, and recall his velocity found above after he finishes accelerating:

The reason the value  is used instead of  is because he only travels for one second in the constant velocity condition.

A position is function is given by the integral of the velocity function with respect to time:

The constant of integration can be found by utilizing the initial position:

And so

Position is given by the integral of the velocity function with respect to time:

To find the constant of integration, use the initial condition:

This yields the function:

Therefore

Using calculus, you know that you can compute this by "working backword" from the acceleration function.  We know that:

.  Since our initial velocity is , it is:

Now, from this, we can compute the position of the particle by computing:

Since we know that the initial height of the ball was , we likewise know:

Thus, at , we have:

Given the acceleration, the position is the second integral of the function. To find the velocity function, the first integral, find the antiderivative of the acceleration function.

This gives

 where C is a constant that can be found using other information from the problem.

We know that at  the velocity is 0 m/s.

Plugging that in we get 

 and we get .  

Therefore, 

.

To find the position function, take the antiderivative again.

 where D is another constant. Using a similar strategy to find the value of the constant we use that at  the position is  m.  and .  

Finally, we get 

.  

In order to solve this problem, we must realize that the integral of the acceleration function is the veloctiy function and the integral of the velocity function is the position function. Thefore, by taking the double integral of the acceleration function, we can find the position function and therefore the position of the space ship can be found.

The general rule of integration  must be known in order to take the integral of the function.  After taking the integral, it is found that the velocity function is ; taking the integral of the veloctiy function, we find that the position function is .  

Setting the position function equal to 100 km, we solve for .

Start with the function for acceleration, . The anti-derivative is the function for velocity. We are told from the problem that the initial velocity, , is 10 feet per second, so the function for velocity is .

The anti-derivative of this is the function for position. Since the ping-pong ball is at an initial height of 100 feet, the function for position is

. 

The problem asks us to determine how many seconds have passed when the ping-pong ball hits the ground. In other words, what is t when the position s(t) is 0. The easiest way to determine what t is when this function is 0 is to use the quadratic formula:

Of the 2 answers, the only one that makes sense is the positive one - negative time would mean the object hit the ground before it was even tossed!

The function for the leaf's acceleration is . The function for its velocity is the anti-derivative, with a constant of 0 since the initial velocity is 0. This means the function for the leaf's velocity is . The function for the leaf's position is the anti-derivative of that with a constant of 9, since the initial position is 9 feet in the air. This function is then , or .

If we're looking for the leaf to land at Gretel's feet, we want to know when it hits the ground, or in other words what t equals when s(t) is 0: 

subtract 9 from both sides 

divide by -4.9

take the square root

This means that it will take 1.355 seconds for the leaf to hit the ground. If Gretel is moving at 1 meter per second, in 1.355 seconds she could travel exactly 1.355 meters.

You can derive the equations of motion from the given value of acceleration, by integration. Or by memorization.

We want to know the maximum height, but so far have two variables, time and final position.

Set     in order to calculate time.

Now use the quadratic formula to obtain two roots, and use the positive number moving forward. If you don't remember the formula, it is

.

Doing this allows you to calculate the total time that it takes for the ball to reach zero height again (hits the ground - trip is over).

Now divide that number in half to get the midway point in the trip. This will be the time it takes to reach maximum height.

And finally plug this number for time into the equation, you get, 

Now,