# 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:

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

1     ${\displaystyle \Delta \ell =|{\vec {r}}_{2}-{\vec {r}}_{1}|}$, and ${\displaystyle \Delta v=|{\vec {v}}_{2}-{\vec {v}}_{1}|}$

2     ${\displaystyle \Delta \ell =v\Delta t}$ (rate times time equals distance).

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

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

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

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

7     ${\displaystyle a={\frac {v^{2}}{r}}}$

## Practice quiz

all of these questions refer to this image:

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

 Yes No

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

 Yes No

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

 Yes No

4 Is ${\displaystyle adt/v=vdt/r}$ valid for uniform circular motion?

 Yes No

5 Is ${\displaystyle dv=adt}$ valid for uniform circular motion?

 Yes No

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

 Yes No

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

 Yes No

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

 Yes No

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

 Yes No

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

 Yes No

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

 Yes No

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

 Yes No

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

 Yes No

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

 Yes No