Calculus 1 : Solving for Time
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Example Questions
Example Question #31 : Solving For Time
A particle is accelerating at a rate given by the expression 
Find an expression in terms of t that gives the amount of time in seconds it takes for the particle to reach the wall after passing the gate.
The acceleration of the particle is given by the follwoing formula: 
When taking an integral, you can seperate the integral into the sum of the integral of each part.
Therefore, the expression for the velocity of the particle is determined to be:
In order to find the constant C, you must use the conditions provided to you. The radar detects the velocity of the particle to be equal to 50 m/s two seconds after the particle passes the gate. Use this information to find the constant C.
Now that the expression for velocity has been determined, the position function can be found by integrating the velocity function. The position function is used to find how long it takes for the particle to travel 7000 meters.
Once again, this integral can be seperated into a sum of the integral of each part:
Since the start of the 7000 meters begins at the gate, the initial position of the particle is zero. Therefore, the constant C is equal to zero. This gives a position function for the particle:
By setting this equal to the total distance traveled, the time t that it takes to reach 7000 meters can be determined.
or
Example Question #32 : Solving For Time
A particle's velocity is modeled (in feet per second) by the equation:
At 
At what time is the particle's acceleration equal to 578 ft/s/s?
For this question, we must first determine what the accleration function is by taking the derivative of the velocity equation:
Recall the power rule to differentiate.
The power rule states,
The equation for the acceleration of the particle, then, is:
With this equation in hand, we just have to solve for t when 
We are only interested in the positive solution to this problem, since the particle is not in motion before t=0.
Example Question #33 : Solving For Time
Find how long it takes for a projectile starting at 
To solve this, we have to take the first derivative and set it equal to zero:
Remember that derivative of 
Moving 
We now get that:
From here, we can see that this equation will never equal zero since
The correct answer is
Example Question #34 : Solving For Time
There are equations in physics that can tell you about the position and velocity of an object as a function of time. If we were to graph the equation of the relationship between the object's height and time it would be a quadratic while graphing the velocity vs time would be linear (since velocity is the derivative of position in respect to time. The equation of the height of the projectile with respect to time is the following
Say you throw a ball straight up in the air at an initial velocity of 20m/s. At approximately what time does the ball reach it's highest height? Use 
Not enough information given.
Given that the relationship between height and time is a parabola, we note that the greatest height will occur when the first derivative is zero.
Given the equation,
The derivative is
which we then set equal to zero and plug in our known values.
We get the following:
Then solving for time, t, gives us 2.04s.
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