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4 years ago

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Sphere 1 has surface area A₁ and volume V₁, sphere 2 has surface area A₂ and volume V₂. If the radius of sphere 2 is six times t

he radius of sphere 1, what is the ratio $\frac{A2}{A1}$ of the areas? What is the ratio $\frac{V2}{V1}$ of the volumes?

1 answer:

Sveta_85 [38]4 years ago

8 0

Let , radius of sphere 1 is r .

So , radius of sphere 2 is 6r .

Surface area of sphere is given by :

$A=4\pi r^2$

So ,

$\dfrac{A_2}{A_1}=\dfrac{4\pi(6r)^2}{4\pi r^2}\\\\\dfrac{A_2}{A_1}=36$

Volume is given by :

$V=\dfrac{4}{3}\pi r^3$

Ratio of sphere 2 by sphere 1 is given by :

$\dfrac{V_2}{V_1}=\dfrac{\dfrac{4}{3}\pi (6r)^3}{\dfrac{4}{3}\pi r^3}\\\\\dfrac{V_2}{V_1}=216$

Therefore , the ratio of area and volume is 36 and 216 respectively .

Hence , this is the required solution .

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Answer:

Immediately after throwing the backpack away, twin A would be moving away from twin B at approximately $0.630\; \rm m \cdot s^{-1}$ .

Initially, twin B would not immediately be moving. However, after the backpack hits her, she would move away from twin A at approximately $0.526\; \rm m \cdot s^{-1}$ if she held onto the backpack.

Explanation:

Consider this scenario in three steps:

Step one: twin A is carrying the backpack.
Step two: twin A throws the backpack away; the backpack is en route to twin B;
Step three: twin B starts to move after the backpack hits her.

Since all external forces are ignored, momentum should be conserved when changing from step one to step two, and from step two to step three.

<h3>From step one to step two</h3>

In step one, neither twin A nor the backpack is moving. Their initial momentum would be zero. That is:

$p(\text{twin A, step one}) = 0$ .
$p(\text{backpack, step one}) = 0$ .

Therefore:

$p(\text{backpack, step one}) +p(\text{twin A, step one}) = 0$ .

In step two, the backpack is moving towards twin B at $3.20\; \rm m \cdot s^{-1}$ . Since the mass of the backpack is $12.0\; \rm kg$ , its momentum at that point would be:

$\begin{aligned}p(\text{backpack, step two}) &= m \cdot v \\ &= 12.0\;\rm kg \times 3.20\; \rm m \cdot s^{-1} = 38.4\; \rm kg\cdot m \cdot s^{-1} \end{aligned}$ .

Momentum is conserved when twin A throws the backpack away. Hence:

$\begin{aligned}&p(\text{backpack, step two}) +p(\text{twin A, step two}) \\ &= p(\text{backpack, step one}) +p(\text{twin A, step one})\end{aligned}$ .

Therefore:

$p(\text{twin A, step two}) \\ &= p(\text{backpack, step one}) +p(\text{twin A, step one}) - p(\text{backpack, step two}) \\ &= -38.4\; \rm kg \cdot m \cdot s^{-1}\end{aligned}$ .

The mass of twin A (without the backpack) is $61.0\; \rm kg$ . Therefore, her velocity in step two would be:

$\begin{aligned} v(\text{twin A, step two}) &= \frac{p}{m} \\ &= \frac{-38.4\; \rm kg \cdot m \cdot s^{-1}}{61.0\; \rm kg} \approx -0.630\; \rm m \cdot s^{-1}\end{aligned}$ .

Note that while the velocity of the backpack is assumed to be greater than zero, the velocity of twin A here is less than zero. Since the backpack is moving towards twin B, it can be concluded that twin A is moving in the opposite direction away from twin B.

<h3>From step two to step three</h3>

In step two:

$p(\text{twin B, step two}) = 0$ since twin B is not yet moving.
$p(\text{backpack, step two}) = 38.4\; \rm kg \cdot m\cdot s^{-1}$ from previous calculations.

Assume that twin B holds onto the incoming backpack. Thus, the velocity of the backpack and twin B in step three will be the same. Let $v(\text{twin B and backpack, step three})$ denote that velocity.

In step three, the sum of the momentum of twin B and the backpack would thus be:

$\begin{aligned}& m(\text{twin B}) \cdot v(\text{twin B and backpack, step three}) \\ &+ m(\text{backpack}) \cdot v(\text{twin B and backpack, step three})\end{aligned}$ .

Simplify to obtain:

$(m(\text{twin B}) + m(\text{backpack})) \cdot v(\text{twin B and backpack, step three})$ .

Momentum is conserved when twin B receives the backpack. Therefore:

$\begin{aligned}& (m(\text{twin B}) + m(\text{backpack})) \cdot v(\text{twin B and backpack, step three})\\ =&p(\text{twin B, step two}) +p(\text{backpack, step two})\\ =& 38.4\; \rm kg \cdot m\cdot s^{-1} \end{aligned}$ .

Therefore:

$\begin{aligned}& v(\text{twin B and backpack, step three})\\ =&\frac{p(\text{twin B, step two}) +p(\text{backpack, step two})}{m(\text{twin B}) + m(\text{backpack})}\\ =& \frac{38.4\; \rm kg \cdot m\cdot s^{-1}}{61.0\; \rm kg + 12.0\; \rm kg} \approx 0.526\;\rm m\cdot s^{-1} \rm \end{aligned}$ .

In other words, if twin B holds onto the backpack, then (after doing so) she would be moving away from twin A at approximately $0.526\; \rm m \cdot s^{-1}$ .

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