Deriving Speed In Terms of Potential Difference

[Oriako]

I'm attempting to derive speed in terms of potential difference (voltage), can anyone let me know if there are any errors?

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21E_k%3D%5Cfrac%7B1%7D%7B2%7D%5Ctext%7Bmv%7D%5E2.gif

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21v%3D%5Csqrt%7B%5Cfrac%7B2E_p%7D%7Bm%7D%7D.gif, since [PLAIN]http://www.texify.com/img/%5CLARGE%5C%21E_k%3DE_p.gif

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21v%3D%5Csqrt%7B%5Cfrac%7B2%28%5Ctext%7Bma%24%5CDelta%20%24d%7D%29%7D%7Bm%7D%7D.gif, since [PLAIN]http://www.texify.com/img/%5CLARGE%5C%21E_p%3D%5Ctext%7Bmg%24%5CDelta%20%24h%7D%20.gif may be generalized to [PLAIN]http://www.texify.com/img/%5CLARGE%5C%21E_p%3D%5Ctext%7Bma%24%5CDelta%20%24d%7D.gif for electric energy.

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21v%3D%5Csqrt%7B%5Cfrac%7B2%5Cleft%28F_%7B%5Ctext%7Bnet%7D%7D%5Ctext%7B%24%5CDelta%20%24d%7D%5Cright%29%7D%7Bm%7D%7D.gif, since [PLAIN]http://www.texify.com/img/%5CLARGE%5C%21F_%7B%5Ctext%7Bnet%7D%7D%3D%5Ctext%7Bma%7D.gif

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21v%3D%5Csqrt%7B%5Cfrac%7B2W%7D%7Bm%7D%7D.gif, since [PLAIN]http://www.texify.com/img/%5CLARGE%5C%21W%3DF_%7B%5Ctext%7Bnet%7D%7D%5Ctext%7B%24%5CDelta%20%24d%7D.gif

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21v%3D%5Csqrt%7B%5Cfrac%7B2%5Ctext%7B%24%5CDelta%20%24E%7D%7D%7Bm%7D%7D.gif, since [PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Ctext%7B%24%5CDelta%20%24E%7D%20%3D%20W.gif

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21v%3D%5Csqrt%7B%5Cfrac%7B2%28%5Ctext%7B%24%5CDelta%20%24Vq%7D%29%7D%7Bm%7D%7D.gif, since [PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Ctext%7B%24%5CDelta%20%24Vq%7D%3D%5Ctext%7B%24%5CDelta%20%24E%7D.gif

Therefore, speed is equal to the square root of double the product of charge and potential difference divided by mass.

Thanks!

 

[Doc Al]

Seems like that took a lot of work! If you accept conservation of energy, thus ΔKE = qΔV (ignoring signs), then you can get to the punchline in one step.

 

[Oriako]

Sorry, forgot to mention it. But I did this to prove to myself that ΔE = $$E_k$$