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

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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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uniform circular motion (here the Latin d was used instead of the Greek Δ

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:

Point added for a correct answer:
Points for an incorrect answer:
Ignore the questions' coefficients:

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