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In summary: Solve for Vi:Vi = (D - Di + 1/2 a t2) / ta = 2 (D - Di - Vi t) / t2Vi = -a t + Vit = (Vi - Vf) / a = (Vf - Vi) / aIn summary, the formulas for initial velocity (Vi), acceleration (a), and time (t) in the given conversation are derived from the formula D = Vit + 1/2at2 and can be solved using algebra and the concept of constant acceleration. However, to fully understand and derive these formulas, some knowledge of calculus is necessary.
- #1
Leoragon
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I'm trying to understand this formula but I'm stuck. I've looked up how to obtain the initial velocity, the acceleration, and time. But all I see are formulas, I don't know how they got it.
Can someone guide me through this please?
D = Vit + 1/2at2
Vi = (d - 1/2at2) / t
a = 2 x (d - Vit) / t2
t = -Vi / a [itex]\pm[/itex] (√[\(Vi)2 + 2ad]) / a
http://http://answers.yahoo.com/question/index?qid=20090621164821AAHxOMD Here's my reference.
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- #2
MarneMath
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Well, do you know calculus? If so, consider integration of some constant acceleration a. Then the rest is algebra. For example, your V_i = (d - 1/2at^2)/t is just the first equation but solved for V_i.
- #3
Leoragon
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MarneMath said:
Nope, I don't. Can you get these formulas without it?
- #4
MarneMath
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Well, yes and no. Once you find one equation, you can find the others by algebra. However, I don't know a way of finding one without, at the very least, using ideas from Calculus. *
*At the same time, I've never tried to find such a way, so maybe there exist a way that I'm not aware of.
- #5
Leoragon
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So these are the right formulas? But where did they come from? What do they mean?
- #6
chingel
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If you start with some speed and have a constant acceleration a, after time t your speed will be at. Since the acceleration was constant, the average speed is just the average of the starting and finishing speeds. Then the distance is just the average speed times time. Try to work it out, find the average speed times t and see what you get.
- #7
HallsofIvy
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Because you are talking about "instantaneous" speed and acceleration, and those are concepts that cannot even be defined without appealing to some kind of Calculus or limit process, you are not going to be able to derive these equations without Calculus. As chigel says, with constant acceleration, you can average speed over an integral to be the arithetic average of the beginning and ending speeds, but showing that requires Calculus (specifically, the "mean value theorem").
Leoragon
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Well I mean average velocity or acceleration. How can I find these variables using algebra if I already have the first formula (s = ut X 1/2at^2)?
- #9
Millennial
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If you are referring to weighted averages, the definition of the weighted average of a data set d is given by [itex]\displaystyle \frac{\sum w_k d_k}{\sum w_k}[/itex] where w denotes the weight of each data. Taking the data set to be the velocity and the weight to be the time, one can solve a question like this:
"A car moves x distance with velocity 20 and 4x distance with velocity 40. What is the average velocity of the car?"
To solve this, we simply note that the car traveled t time with velocity 20 and 2t time with velocity 40, and hence we have the average as [itex]\displaystyle \frac{20t+80t}{t+2t}=\frac{100}{3}\approx 33.3[/itex].
However, the derivation of the weighted average requires the mean value theorem, so I do not know how useful you might find this to be.
- #10
rcgldr
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This can be done without calculus as long as you accept that with constant acceleration the average velocity Va = is 1/2 (Vi + Vf), where Vi is initial velocity and Vf is final velocity.
Va = 1/2 (Vi + Vf)
Vf = Vi + a t
Va = 1/2 (Vi + (Vi + a t)) = Vi + 1/2 a t
D = Di + Va t = Di + (Vi + 1/2 a t) t = Di + Vi t + 1/2 a t2
Related to Understanding the D = Vit + 1/2at^2 Formula | Tips and Tricks
1. What does the equation D = Vit + 1/2at^2 represent?
The equation D = Vit + 1/2at^2 represents the displacement (D) of an object over time (t) under constant acceleration (a) starting at an initial velocity (Vi).
2. How is displacement (D) calculated using this equation?
To calculate displacement (D) using this equation, you need to know the initial velocity (Vi) of the object, the acceleration (a) it is experiencing, and the time (t) it has been moving. Plug these values into the equation D = Vit + 1/2at^2 and solve for D.
3. What is the significance of the 1/2 factor in the equation?
The 1/2 factor in the equation represents the acceleration due to gravity (g) being divided by 2. This is necessary because the object experiences this acceleration over the entire time (t) it is in motion, so it needs to be divided by 2 to account for the average acceleration over that time period.
4. Can this equation be used for any type of motion?
Yes, this equation can be used for any type of motion as long as the acceleration (a) is constant. This means that the acceleration does not change throughout the motion of the object.
5. How does this equation relate to the laws of motion?
This equation is derived from the second law of motion, which states that the net force (F) applied to an object is equal to its mass (m) multiplied by its acceleration (a). By rearranging this equation to solve for acceleration (a), we get a = F/m. When this is plugged into the equation D = Vit + 1/2at^2, we get the final form of the equation that relates displacement (D) to the initial velocity (Vi), time (t), and acceleration (a).
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