# Physics equations/06-Uniform Circular Motion and Gravitation/Q:derive

The following derivation of a=v2/r for uniform circular motion needs to be studied before the student attempts this quiz:

click to view or hide proof

Using the figure we define the distance traveled by a particle during a brief time interval, $\Delta t$ , and the (vector) change in velocity:

1     $\Delta \ell =|{\vec {r}}_{2}-{\vec {r}}_{1}|$ , and $\Delta v=|{\vec {v}}_{2}-{\vec {v}}_{1}|$ 2     $\Delta \ell =v\Delta t$ (rate times time equals distance).

3     $\Delta {\vec {v}}={\vec {a}}\Delta t$ (definition of acceleration).

4     $\Delta v=a\Delta t$ (taking the absolute value of both sides).

5     ${\frac {\Delta v}{v}}={\frac {\Delta \ell }{r}}$ (by similar triangles). Substituting (2) and (4) yields:

6     ${\frac {a\Delta t}{v}}={\frac {v\Delta t}{r}}$ , which leads to ${\frac {a}{v}}={\frac {v}{r}}$ , and therefore:

7     $a={\frac {v^{2}}{r}}$ ## Practice quiz

all of these questions refer to this image:

1 Is $dv/d\ell =v/r$ valid for uniform circular motion?

 Yes No

2 Is $dv=|{\vec {v}}_{2}-{\vec {v}}_{1}|$ valid for uniform circular motion?

 Yes No

3 Is $d\ell =vdt$ valid for uniform circular motion?

 Yes No

4 Is $adt/v=vdt/r$ valid for uniform circular motion?

 Yes No

5 Is $dv=adt$ valid for uniform circular motion?

 Yes No

6 Is $|d{\vec {v}}|=adt$ valid for uniform circular motion?

 Yes No

7 Is $d\ell =|{\vec {r}}_{2}-{\vec {r}}_{1}|$ valid for uniform circular motion?

 Yes No

8 Is $d\ell =|{\vec {r}}_{2}|-|{\vec {r}}_{1}|$ valid for uniform circular motion?

 Yes No

9 Is $v/d\ell =r/dv$ valid for uniform circular motion?

 Yes No

10 Is $dv/r=d\ell /v$ valid for uniform circular motion?

 Yes No

11 Is $rd\ell =vdv$ valid for uniform circular motion?

 Yes No

12 Is $dv=|{\vec {v}}_{2}|-|{\vec {v}}_{1}|$ valid for uniform circular motion?

 Yes No

13 Is $d\ell /dv=v/r$ valid for uniform circular motion?

 Yes No

14 Is $dv/d\ell =r/v$ valid for uniform circular motion?

 Yes No