Finding specific antiderivatives using initial conditions, including applications to motion along a line - AP Calculus AB
Card 1 of 99
A particle is moving in a straight path with a constant initial velocity. The particle is then subjected to a force causing a time-dependent acceleration given as a function of time:
After 10 seconds, the particle has a velocity equal to 
meters-per-second. Find the initial velocity in terms of the constants 
, 
and 
Units are all in S.I. (meters, seconds, meters-per-second, etc.)
A particle is moving in a straight path with a constant initial velocity. The particle is then subjected to a force causing a time-dependent acceleration given as a function of time:
After 10 seconds, the particle has a velocity equal to
Units are all in S.I. (meters, seconds, meters-per-second, etc.)
Tap to reveal answer
Begin by finding the velocity function by integrating the acceleration function.
We use 
as the constant of integration since the function 
is a velocity and at initially, at 
, the velocity is equals the constant of integration.
At 
seconds we are told the velocity is equal to 
.
Begin by finding the velocity function by integrating the acceleration function.
We use
At
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Find the average value of the function 
on the interval 
Find the average value of the function
Tap to reveal answer
The average value of a function on a given interval 
is given by the following function:
Now, let's simply input our values and function in:
The average value of a function on a given interval
Now, let's simply input our values and function in:
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Determine the position function for a particle whose velocity is given by the equation
and whose initial position is 10.
Determine the position function for a particle whose velocity is given by the equation
and whose initial position is 10.
Tap to reveal answer
The position function describing any object is the antiderivative of the velocity function (in other words, velocity is the derivative of position).
So, we first integrate the velocity function:
The following rule was used for integration:
Now, to determine the constant of integration, we use our initial condition given,
Plugging this into our function, we get
Our final answer is
The position function describing any object is the antiderivative of the velocity function (in other words, velocity is the derivative of position).
So, we first integrate the velocity function:
The following rule was used for integration:
Now, to determine the constant of integration, we use our initial condition given,
Plugging this into our function, we get
Our final answer is
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The function describing the acceleration of a spacecraft with respect to time is
Determine the function describing the position of a spacecraft given that the initial acceleration is 0, the initial velocity is 3, and the initial position is 9.
The function describing the acceleration of a spacecraft with respect to time is
Determine the function describing the position of a spacecraft given that the initial acceleration is 0, the initial velocity is 3, and the initial position is 9.
Tap to reveal answer
To find the position function from the acceleration function, we integrate the acceleration function to find the velocity function, and integrate again to get the position function:
The integral was found using the following rule:
To find the constant of integration, we use the initial velocity condition given:
Now, after replacing C with the known value, we integrate the velocity function to get the position function:
The same rule of integration was used as above.
We use the same procedure to solve for C, too, only this time using the initial position condition:
Our final answer is
To find the position function from the acceleration function, we integrate the acceleration function to find the velocity function, and integrate again to get the position function:
The integral was found using the following rule:
To find the constant of integration, we use the initial velocity condition given:
Now, after replacing C with the known value, we integrate the velocity function to get the position function:
The same rule of integration was used as above.
We use the same procedure to solve for C, too, only this time using the initial position condition:
Our final answer is
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Given a particle with an acceleration at time 
to be 
. With initial conditions 
and 
where 
is the velocity at time 
, and 
is position of the particle at time 
.
Find the position at time 
.
Given a particle with an acceleration at time
Find the position at time
Tap to reveal answer
We first must establish the following relationship
and 
We now may note that
or 
Since 
We must plug in our initial condition 
Therefore our new velocity equation is 
We now may similarly integrate the velocity equation to find position.
Plugging in our second initial condition 
We find our final equation to be:
We first must establish the following relationship
We now may note that
Since
We must plug in our initial condition
Therefore our new velocity equation is
We now may similarly integrate the velocity equation to find position.
Plugging in our second initial condition
We find our final equation to be:
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Find the integral which satisfies the specific conditions of 
Find the integral which satisfies the specific conditions of
Tap to reveal answer
Find the integral which satisfies the specific conditions of 
To do this problem, we need to recall that integrals are also called anti-derivatives. This means that we can calculate integrals by reversing our integration rules.
Furthermore, to find the specific answer using initial conditions, we need to find our "c" at the end.
Thus, we can have the following rules.
Using these rules, we can find our answer:
Will become:
And so our anti-derivative is:
Now, let's find c. First set our above expression equal to y
Next, plug in 
for y and t. Then solve for c
Looks a bit messy, but we can clean it up to get:
Now, to solve, simply replace c with 12.12
Find the integral which satisfies the specific conditions of
To do this problem, we need to recall that integrals are also called anti-derivatives. This means that we can calculate integrals by reversing our integration rules.
Furthermore, to find the specific answer using initial conditions, we need to find our "c" at the end.
Thus, we can have the following rules.
Using these rules, we can find our answer:
Will become:
And so our anti-derivative is:
Now, let's find c. First set our above expression equal to y
Next, plug in
Looks a bit messy, but we can clean it up to get:
Now, to solve, simply replace c with 12.12
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Find the velocity function given the following information:
The acceleration function is 
;
Find the velocity function given the following information:
The acceleration function is
Tap to reveal answer
To find the velocity function, we integrate the acceleration function (the acceleration is the antiderivative of the velocity):
The rules of integration used were
, 
To solve for the integration constant, we plug in the given initial condition:
Our final answer is
To find the velocity function, we integrate the acceleration function (the acceleration is the antiderivative of the velocity):
The rules of integration used were
To solve for the integration constant, we plug in the given initial condition:
Our final answer is
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What is the position function if the initial position is 0 and the velocity function is given by 
?
What is the position function if the initial position is 0 and the velocity function is given by
Tap to reveal answer
To find the position function, we must integrate the velocity function, as velocity is the antiderivative of position:
The following rule of integration was used:
Finally, we use the initial condition to solve for the integration constant:
Our final answer is
To find the position function, we must integrate the velocity function, as velocity is the antiderivative of position:
The following rule of integration was used:
Finally, we use the initial condition to solve for the integration constant:
Our final answer is
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A particle at the origin has an initial velocity of 
. If its acceleration is given by 
, find the position of the particle after 1 second.
A particle at the origin has an initial velocity of
Tap to reveal answer
In this problem, letting 
denote the position of the particle and 
denote the velocity, we know that 
. Integrating and working backwards we have,
Plugging in our initial condition, 
, we see immediately that 
.
Repeating the process again for 
, we find that
Plugging in our initial condition, 
(we started at the origin) we see that 
. This gives us a final equation
. The problem asks for 
which is simply 
In this problem, letting
Plugging in our initial condition,
Repeating the process again for
Plugging in our initial condition,
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A particle is moving in a straight path with a constant initial velocity. The particle is then subjected to a force causing a time-dependent acceleration given as a function of time:
After 10 seconds, the particle has a velocity equal to meters-per-second. Find the initial velocity in terms of the constants , and
Units are all in S.I. (meters, seconds, meters-per-second, etc.)
A particle is moving in a straight path with a constant initial velocity. The particle is then subjected to a force causing a time-dependent acceleration given as a function of time:
After 10 seconds, the particle has a velocity equal to
Units are all in S.I. (meters, seconds, meters-per-second, etc.)
Tap to reveal answer
Begin by finding the velocity function by integrating the acceleration function.
We use as the constant of integration since the function is a velocity and at initially, at , the velocity is equals the constant of integration.
At seconds we are told the velocity is equal to .
Begin by finding the velocity function by integrating the acceleration function.
We use
At
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Find the average value of the function on the interval
Find the average value of the function
Tap to reveal answer
The average value of a function on a given interval is given by the following function:
Now, let's simply input our values and function in:
The average value of a function on a given interval
Now, let's simply input our values and function in:
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Determine the position function for a particle whose velocity is given by the equation
and whose initial position is 10.
Determine the position function for a particle whose velocity is given by the equation
and whose initial position is 10.
Tap to reveal answer
The position function describing any object is the antiderivative of the velocity function (in other words, velocity is the derivative of position).
So, we first integrate the velocity function:
The following rule was used for integration:
Now, to determine the constant of integration, we use our initial condition given,
Plugging this into our function, we get
Our final answer is
The position function describing any object is the antiderivative of the velocity function (in other words, velocity is the derivative of position).
So, we first integrate the velocity function:
The following rule was used for integration:
Now, to determine the constant of integration, we use our initial condition given,
Plugging this into our function, we get
Our final answer is
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The function describing the acceleration of a spacecraft with respect to time is
Determine the function describing the position of a spacecraft given that the initial acceleration is 0, the initial velocity is 3, and the initial position is 9.
The function describing the acceleration of a spacecraft with respect to time is
Determine the function describing the position of a spacecraft given that the initial acceleration is 0, the initial velocity is 3, and the initial position is 9.
Tap to reveal answer
To find the position function from the acceleration function, we integrate the acceleration function to find the velocity function, and integrate again to get the position function:
The integral was found using the following rule:
To find the constant of integration, we use the initial velocity condition given:
Now, after replacing C with the known value, we integrate the velocity function to get the position function:
The same rule of integration was used as above.
We use the same procedure to solve for C, too, only this time using the initial position condition:
Our final answer is
To find the position function from the acceleration function, we integrate the acceleration function to find the velocity function, and integrate again to get the position function:
The integral was found using the following rule:
To find the constant of integration, we use the initial velocity condition given:
Now, after replacing C with the known value, we integrate the velocity function to get the position function:
The same rule of integration was used as above.
We use the same procedure to solve for C, too, only this time using the initial position condition:
Our final answer is
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Given a particle with an acceleration at time to be . With initial conditions and where is the velocity at time , and is position of the particle at time .
Find the position at time .
Given a particle with an acceleration at time
Find the position at time
Tap to reveal answer
We first must establish the following relationship
and
We now may note that
or
Since
We must plug in our initial condition
Therefore our new velocity equation is
We now may similarly integrate the velocity equation to find position.
Plugging in our second initial condition
We find our final equation to be:
We first must establish the following relationship
We now may note that
Since
We must plug in our initial condition
Therefore our new velocity equation is
We now may similarly integrate the velocity equation to find position.
Plugging in our second initial condition
We find our final equation to be:
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Find the integral which satisfies the specific conditions of
Find the integral which satisfies the specific conditions of
Tap to reveal answer
Find the integral which satisfies the specific conditions of
To do this problem, we need to recall that integrals are also called anti-derivatives. This means that we can calculate integrals by reversing our integration rules.
Furthermore, to find the specific answer using initial conditions, we need to find our "c" at the end.
Thus, we can have the following rules.
Using these rules, we can find our answer:
Will become:
And so our anti-derivative is:
Now, let's find c. First set our above expression equal to y
Next, plug in for y and t. Then solve for c
Looks a bit messy, but we can clean it up to get:
Now, to solve, simply replace c with 12.12
Find the integral which satisfies the specific conditions of
To do this problem, we need to recall that integrals are also called anti-derivatives. This means that we can calculate integrals by reversing our integration rules.
Furthermore, to find the specific answer using initial conditions, we need to find our "c" at the end.
Thus, we can have the following rules.
Using these rules, we can find our answer:
Will become:
And so our anti-derivative is:
Now, let's find c. First set our above expression equal to y
Next, plug in
Looks a bit messy, but we can clean it up to get:
Now, to solve, simply replace c with 12.12
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Find the velocity function given the following information:
The acceleration function is ;
Find the velocity function given the following information:
The acceleration function is
Tap to reveal answer
To find the velocity function, we integrate the acceleration function (the acceleration is the antiderivative of the velocity):
The rules of integration used were
,
To solve for the integration constant, we plug in the given initial condition:
Our final answer is
To find the velocity function, we integrate the acceleration function (the acceleration is the antiderivative of the velocity):
The rules of integration used were
To solve for the integration constant, we plug in the given initial condition:
Our final answer is
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What is the position function if the initial position is 0 and the velocity function is given by ?
What is the position function if the initial position is 0 and the velocity function is given by
Tap to reveal answer
To find the position function, we must integrate the velocity function, as velocity is the antiderivative of position:
The following rule of integration was used:
Finally, we use the initial condition to solve for the integration constant:
Our final answer is
To find the position function, we must integrate the velocity function, as velocity is the antiderivative of position:
The following rule of integration was used:
Finally, we use the initial condition to solve for the integration constant:
Our final answer is
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A particle at the origin has an initial velocity of . If its acceleration is given by , find the position of the particle after 1 second.
A particle at the origin has an initial velocity of
Tap to reveal answer
In this problem, letting denote the position of the particle and denote the velocity, we know that . Integrating and working backwards we have,
Plugging in our initial condition, , we see immediately that .
Repeating the process again for , we find that
Plugging in our initial condition, (we started at the origin) we see that . This gives us a final equation
. The problem asks for which is simply
In this problem, letting
Plugging in our initial condition,
Repeating the process again for
Plugging in our initial condition,
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A particle is moving in a straight path with a constant initial velocity. The particle is then subjected to a force causing a time-dependent acceleration given as a function of time:
After 10 seconds, the particle has a velocity equal to meters-per-second. Find the initial velocity in terms of the constants , and
Units are all in S.I. (meters, seconds, meters-per-second, etc.)
A particle is moving in a straight path with a constant initial velocity. The particle is then subjected to a force causing a time-dependent acceleration given as a function of time:
After 10 seconds, the particle has a velocity equal to
Units are all in S.I. (meters, seconds, meters-per-second, etc.)
Tap to reveal answer
Begin by finding the velocity function by integrating the acceleration function.
We use as the constant of integration since the function is a velocity and at initially, at , the velocity is equals the constant of integration.
At seconds we are told the velocity is equal to .
Begin by finding the velocity function by integrating the acceleration function.
We use
At
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Find the average value of the function on the interval
Find the average value of the function
Tap to reveal answer
The average value of a function on a given interval is given by the following function:
Now, let's simply input our values and function in:
The average value of a function on a given interval
Now, let's simply input our values and function in:
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