Calculus 1 : Position
Study concepts, example questions & explanations for Calculus 1
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Example Questions
Example Question #171 : How To Find Position
Determine the position of a runner 5 seconds in the future who's accelerating constantly at a rate of 
The formula for position given acceleration, initial velocity, and position is:
We can describe the acceleration as:
At 
Example Question #1022 : Spatial Calculus
If v(t) represents the velocity of a water balloon dropped off a sky scraper, find the function which represents its position if 
If v(t) represents the velocity of a water balloon dropped off a sky scraper, find the function which represents its position if 
We are given a velocity function and asked to find a position function. This means we will have to integrate. (Recall that velocity is the first derivative of position)
So, we need to evaluate the following integral:
To integrate a monomial, simply increase its exponent by one, and divide by the new exponent.
Now, we are almost done, but we need to solve for c. We use our initial conditions to do so.
So our position function is:
Example Question #171 : How To Find Position
If v(t) represents the velocity of a water balloon dropped off a sky scraper, find the balloon's position after 3 seconds, if 
If v(t) represents the velocity of a water balloon dropped off a sky scraper, find the balloon's position after 3 seconds, if 
We are given a velocity function and asked to find a position function. This means we will have to integrate. (Recall that velocity is the first derivative of position)
So, we need to evaluate the following integral:
To integrate a monomial, simply increase its exponent by one, and divide by the new exponent.
Now, we need to solve for c. We use our initial conditions to do so.
So our position function is:
Finally, plug in 3 for t to get our answer:
So after 3 seconds, the balloon is still 202.4 units from the ground.
Example Question #1024 : Spatial Calculus
Given the position function of a projectile, using calculus concepts, at what point in time does the projectile reach its highest altitude? What is this altitude (in meters)?
t=1, 3m
t=1.5, 3m
t=1, 2m
t=1, 7m
t=0.5, 7m
t=1, 7m
To find the highest altitude of a projectile given the position function, we must know where the projectile's direction changes from increasing to decreasing. That is, where the projectile reaches a maximum altitude. This occurs at the point when the velocity is equal to zero.
The velocity of the projectile is modeled by the first derivative of the position function.
The velocity is set equal to zero to solve for t.
The velocity is equal to zero at t=1.
This means that at t=1, the projectile reaches its maximum altitude.
This altitude is solved by solving for s(1).
The highest altitude is 7m and it occurs at t=1.
Example Question #1025 : Spatial Calculus
Find the position of an object at t=5, given its velocity function 
To find the position function, one must first take the integral with respect to t of the velocity function,
Using the rules of integration
where C is some constant and therefore,
Since the initial condition of position is given at zero we can solve for C.
Then, by plugging in 2 for t, the correct answer
is obtained.
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