Integrate to get Position function

In summary, the problem asks for the deceleration needed in order for a car traveling at 30m/s to stop in time to avoid an accident 60 meters ahead. Using the equations v(t)= at+30 and s(t)= (1/2)at^2+30t, we can set v(t) = 0 and s(t) = 60 to solve for a, the deceleration needed.
  • #1
lax1113
179
0

Homework Statement


If a car is going 30m/s and see's an accident 60 meters in front of him/her, at what deceleration must the car apply brakes in order to stop in time?

Homework Equations


s(t) = [tex]\int(v)t[/tex]
v(t) = [tex]\int(a)t[/tex]

The Attempt at a Solution


So I know the physics equation that I could use for this very easily (vf^2 = vo^2+2a \Delta (x)
But, for my calculus class I have to do it obvously with only calculus. I don't know why i can't get this, but I have a feeling it will be on my exam tomorrow so I need to know how to show the work.

I am setting it up that V(t) = at + Vot, then integrating to find s(t), but after this point, I have
s(t) = 1/2 a t^2 + vot +s(0) s(0) is 0 for this case because it is from the point of applying brakes... Anyway, where do I go from here? I am kinda lost and really don't know why.

Any help greatly appreciated, thanks!
 
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  • #2
Okay, since you were initially going at 30 m/s, you have [itex]v(t)= at+ 30[/itex] and [itex]s(t)= (1/2)at^2+ 30t[/itex].

The point is that you want to have v(t)= 0 before s(t)= 60. Since the problem asks for a single answer, you want to stop just in time: v(t)= 0 and s(t)= 60. Solve those two equations for a.
 
  • #3
Halls,
Thank you very much for that. I don't know why I just couldn't see it right, I took physics more recently than calculus, so I couldn't remember exactly how to derive the equations.

By the way, a question very similar to this was on my final and i got it! thanks again!


Ben
 

Related to Integrate to get Position function

What is the concept of integration in physics?

Integration in physics refers to the mathematical process of finding the area under a curve on a graph. In other words, it is a way to calculate the total amount of something, such as displacement or velocity, over a period of time.

What is the position function and how is it related to integration?

The position function, also known as the displacement function, is a mathematical representation of an object's position at a given time. It is directly related to integration as it is the result of integrating the velocity function over a certain time interval.

Can integration be used to find position functions for all types of motion?

Yes, integration can be used to find position functions for all types of motion, whether it is constant, accelerated, or decelerated. However, the specific equation used for integration may vary depending on the type of motion.

How does the initial position and velocity affect the position function?

The initial position and velocity of an object can affect the position function by changing the constants in the equation. For example, if an object starts at a different position or with a different velocity, the constants in the position function will be different, resulting in a different function.

Why is it important to use integration to find the position function?

Using integration to find the position function allows us to accurately track and predict the position of an object over time. This is crucial in many fields of science, such as physics and engineering, where understanding an object's motion and position is essential for solving problems and making predictions.

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