Trouble Deriving a Constant Acceleration Formula

In summary, the formula you attempted to use, V=V0 + at, is incorrect when velocity is not constant. The correct formula is x = x0 + v0 t + 1/2 a t2. This can be derived through integration or by using the concept of average velocity. It is important to understand the correct formula in order to solve problems accurately.
  • #1
berenmacbowma
7
0
Q: How long does it take a car to cross a 30 m-wide intersection after the light turns green, if it accelerates from rest at a constant 2.00 m/s^2?

Attempt: V=V0 + at; x-x0/t=v0+at; x-x0=v0+at^2; x=v0+x0+at^2; 30=0+0+2t^2; 15=t^2.

The square root of 15 ended up being my final answer, when the real answer was closer to 5.48 seconds. I just started physics, so I need to know what I personally did wrong when USING the formula I attempted to use. Giving me a different formula won't be helpful, because it won't tell me what I did wrong, and because my book has a whole chart of formulas I could refer to. If someone could explain to me how I derived that formula incorrectly, it would be a load of help. Thank you!
 
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  • #2
You derived the formula incorrectly by assuming that velocity at time t is displacement/time. That it is not true when velocity is not constant. To compound the problem, your algebra is wrong in equation 3, but it doesn't matter since equation 2 is wrong.The kinematic equations are best derived form the calculus, check out a good website for this. You might want to refer to the chart of formulas in the book, and you will discover that none of them are the same as the one you derived.
 
  • #3
berenmacbowma said:
Q: How long does it take a car to cross a 30 m-wide intersection after the light turns green, if it accelerates from rest at a constant 2.00 m/s^2?

Attempt: V=V0 + at; x-x0/t=v0+at

Right there; that does not follow. The only way you're going to be able to make the correct step is through integration, which if you're not familiar with isn't going to be possible at all.
 
  • #4
You can avoid integration by noting that in a constant acceleration situation, average velocity for any time period equals half the sum of the starting and ending velocities.

average veloctiy = av = 1/2 (v0 + v1)
v1 = v0 + at
av = 1/2 (v0 + (v0 + at))
av = 1/2 (2v0 + at)
av = v0 + 1/2 a t

then distance equals x0 + average velocity x time:

x = x0 + av t = x0 + (v0 + 1/2 a t) t
x = x0 + v0 t + 1/2 a t2
 
  • #5


There are a few things that could have gone wrong in your derivation of the constant acceleration formula. One possibility is that you may have made a mistake in your algebraic manipulation. It can be helpful to double check your work and make sure you are following the correct steps to solve for time.

Another possibility is that you may have used the wrong initial velocity (V0) in your equation. Since the car is starting from rest, the initial velocity should be 0 m/s. Using the wrong initial velocity can lead to a different answer.

Additionally, it is important to pay attention to the units in the equation. In this case, the acceleration is given in meters per second squared (m/s^2), so the time should be in seconds (s). If you use the square root of 15 as your answer, you are using the units of meters instead of seconds, which would result in an incorrect answer.

Lastly, it is always a good idea to check your answer using common sense. In this case, if the car is accelerating at a rate of 2.00 m/s^2, it should take less than 15 seconds to cross a 30 m intersection. Therefore, your answer of 5.48 seconds is more reasonable.

In general, when working with formulas, it is important to carefully follow the steps and pay attention to units to ensure an accurate result. It is also helpful to double check your answer using common sense or by plugging it back into the original problem to see if it makes sense. I hope this helps you in your understanding of the constant acceleration formula.
 

1. What is the formula for constant acceleration?

The formula for constant acceleration is a = (vf - vi) / t, where a is the acceleration, vf is the final velocity, vi is the initial velocity, and t is the time interval.

2. Why is it important to derive a constant acceleration formula?

Deriving a constant acceleration formula allows us to understand and predict the motion of objects under constant acceleration. This information is crucial in many fields, such as physics, engineering, and astronomy.

3. What are the key variables in the constant acceleration formula?

The key variables in the constant acceleration formula are acceleration (a), initial velocity (vi), final velocity (vf), and time (t). These variables are all interconnected and can be used to solve for any missing value in the equation.

4. How is the constant acceleration formula derived?

The constant acceleration formula is derived using the principles of calculus. It involves taking the limit of the average acceleration as the time interval approaches zero, resulting in the instantaneous acceleration at a specific moment in time.

5. Are there any real-world applications of the constant acceleration formula?

Yes, the constant acceleration formula has many real-world applications. For example, it is used in the design of roller coasters, calculating the motion of projectiles, and predicting the movement of objects in free fall due to gravity.

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