Why is my book using different equations to solve for velocity?

In summary: We can calculate this by taking the difference of the two potential energies, which is negative in this case. So, the speed of the electron is greater than the speed of the proton.
  • #1
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I am having a difficult time trying to figure out why my book says that I should solve for v by:
(1) Set the kinetic energy equal to the change in potential energy and solve for v

Then 3 problems later it says solve for v by:
(2)Set ΔK = −ΔU and solve for v

http://img593.imageshack.us/img593/1995/1idz.jpg [Broken]


I just need to know why they are using different equations to solve for v.. this is not making any sense.
 
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  • #2
outxbreak said:
I am having a difficult time trying to figure out why my book says that I should solve for v by:
(1) Set the kinetic energy equal to the change in potential energy and solve for v

Then 3 problems later it says solve for v by:
(2)Set ΔK = −ΔU and solve for v


I just need to know why they are using different equations to solve for v.. this is not making any sense.

They are essentially the same equations. ΔK = −ΔU means that the gain in kinetic energy is equal in magnitude to the decrease in potential energy.

The V (upper case) in problem 21 is the potential difference. The v (lower case) is the velocity of the charged particle.

AM
 
  • #3
Andrew Mason said:
They are essentially the same equations. ΔK = −ΔU means that the gain in kinetic energy is equal in magnitude to the decrease in potential energy.


But how do I know when to use
ΔK = −ΔU
or
ΔK = ΔU

That is my problem. :'(
 
  • #4
The book's being sloppy in the first case. It's really using ##\Delta K = \lvert \Delta U \rvert##. For both the electron and proton, the potential energy decreases, so ##\Delta U < 0##.

The proton rolls down electric potential hills, so Vfinal is less than Vinitial. ΔV is therefore negative, and ΔU = e ΔV is also negative. The electron rolls up electric potential hills, so Vfinal is greater than Vinitial. ΔV is therefore positive. The electron has a negative charges, so ΔU = -e ΔV again is negative. Because ΔU is negative, if you try to use ##\frac{1}{2}mv^2 = \Delta U##, there'd be no real solution for v. You need to use ##\Delta K = \frac{1}{2}mv^2 = -\Delta U##. In other words, you always use ##\Delta K = -\Delta U##.

In electromagnetics, it's pretty easy to make sign mistakes if you try to rely solely on the math. It's often easier simply to neglect the signs and calculate the magnitude of quantities and then insert the correct sign based on physical intuition. For instance, in both cases here, we know that the particles are going to speed up, so their kinetic energy increases. This increase comes at the expense of a decrease in potential energy. To find the speed, we don't really care about the sign of ΔU; we just need its magnitude.
 
  • #5


There could be a few reasons why your book is using different equations to solve for velocity. One possibility is that the book is presenting different methods or approaches to solving for velocity, which can be helpful for understanding the concept from different perspectives. Another reason could be that the equations are being applied to different scenarios or problems, which may require different equations to accurately solve for velocity. It is also possible that the equations are simply different forms of the same equation, just rearranged in different ways for convenience or clarity. Ultimately, the important thing is to understand the underlying principles and concepts behind the equations, rather than just memorizing specific equations. This will allow you to apply the appropriate equation for any given situation and solve for velocity accurately. If you are still having trouble understanding why different equations are being used, it may be helpful to consult with your teacher or a tutor for further clarification.
 

1. Why are there different equations for velocity in my book?

There are different equations for velocity because different situations require different equations. For example, the equations used for constant velocity (when the speed remains the same) are different from those used for acceleration (when the speed changes over time). It is important to use the correct equation for the specific situation you are trying to solve.

2. How do I know which equation to use for velocity?

You can determine which equation to use by considering the variables provided in the problem and the type of motion being described. If the speed is constant, you would use the equation v = d/t (velocity = distance/time). If there is acceleration, you would use the equation v = u + at (velocity = initial velocity + acceleration x time).

3. Can I use any equation for velocity to solve a problem?

No, it is important to use the correct equation for the specific situation you are trying to solve. Using the wrong equation may lead to an incorrect answer. It is also important to pay attention to the units of the variables in the equation and make sure they are consistent with the units given in the problem.

4. Why are there multiple equations for velocity in my book?

There are multiple equations for velocity because different scientists and mathematicians have developed different equations for different situations. Some equations may be more useful or accurate in certain situations, so it is important to have a variety of equations to choose from.

5. Can I use the same equation for velocity in all situations?

No, you cannot use the same equation for velocity in all situations. Each equation is designed for a specific situation and using the wrong equation may lead to an incorrect answer. It is important to carefully read and analyze the problem to determine which equation is most appropriate to use.

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