We'll need to work backwards from the acceleration to find the position. We're given the rocket's acceleration, which is opposed by gravity, so the total acceleration will be

The rocket begins at rest, so 

Initial and final time are given as 

Acceleration is the rate of change in velocity, which is the rate of change in position, so to find the equation for position, we'll ned to integrate and use initial conditions twice. 

To integrate, we use the following rule:

Remember, when integrating we have to add the constant, C, to indicate that there's multiple functions that could be our answer.

Integrating then gives us:

To solve for C, we'll use out initial condition, .

We'll need to integrate again to find position:

using the following rule:

Remember, when integrating we have to add the constant, C, to indicate that there's multiple functions that could be our answer.

Integrating then gives us:

To solve for C, we'll use out initial condition, .

which, evaluated at , gives us

The first step in solving this problem is integrating the velocity function to get your position function. When integrating, remember to raise the exponent by 1 and then also put that result on the denominator: . Now, to figure out what C is, plug in your initial conditions given: . Plug in 4 for C to get your answer: .

First, you need to find the derivative of the function. When taling the derivative, multiply the exponent by the coefficient in front of the x term and then decrease the exponent by 1. Therefore: . Now, plug in 1 for x to get your answer: 11.

To find the position function from the velocity function, you must integrate the velocity. When integrating, raise the exponent by 1 and then put that result on the denominator as well: . Then, plug in your initial conditions to find C: . Therefore, the position function is:

To find the position of the particle at time t=1, we must first find the position function of the particle, which is given by the integral of the velocity function:

The integration was performed using the following rule:

Now, plug in the initial condition to solve for C:

Finally, evaluate the position function at t=1,

For any velocity function, v(t), the position function (r(t) or sometimes p(t)) may be found by taking the indefinite integral of v(t).

Now given the intitial condition , we may solve for our constant "C".

We now have the equation

Evaluating for 

To find the position of the particle, you need to integrate the veloctiy function.  Then, plug in   means the -component of the position, and  refers to the -component of the position, so in the end, we will write the solution in coordinate form

To determine the position function, we must integrate the velocity function:

The integral was found using the following rules:

, 

Now, to solve for C, we must use the initial condition:

Replacing C with the value, we get

To find the position function from the acceleration function, integrate  twice.

When integrating, remember to increase the exponent of the variable by one and then divide the term by the new exponent. Do this for each term.

Solve for .

In order to find the position function from the velocity function we need to find the anti-derivative of the velocity function 

When taking the integral we use the inverse power rule which states

Applying this rule we get

To find the value of the constant  we use the initial condition

which yields

Therefore