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, 

We have to start from acceleration and integrate back to position to determine the initial position at .

Since we know that the velocity at  is , we plug it in to determine the initial velocity

, 

Now that we know the function for velocity, we can integrate it to find position

Since we also know that at  that the position is at , we use it to find initial position at 

, 

Therefore, at t=0, the initial position is 1194m. 

In order to find the position equation, we must integrate the velocity function and then plug in the initial condition to solve for C:

We used the following rules for integration:

, 

Now, plug in t=0 into the position equation and solve for C:

Thus, C=0, and our final answer is

The position function can be found by integrating the velocity function with respect to time:

Velocity is given as:

So the position function is:

The integration of constant is found by using the initial condition provided:

Therefore

Position can be found by integrating velocity with respect to time:

For the velocity function

The position function is:

The constant of integration can be found by using the initial condition:

So the definite integral is:

The position function of an object moving with uniform acceleration is , where  is the inital position of the object,  is the inital velocity of the object and  is the acceleration of the object.

For this problem:

, , 

To find the position of the object after a certain time, we first must find the position function of the object, then solve the position function at that given time.

The position function of an object moving with uniform acceleration is , where  is the inital position of the object,  is the inital velocity of the object and  is the acceleration of the object.

For this problem:

, , , 

After  seconds, the ball has travelled  meters.