Space Time & Motion A.2 Forces & Momentum (id: b711ed9bc)

本题目来源于试卷: Space Time & Motion A.2 Forces & Momentum，类别为 IB物理学

[单选题]
A rocket is moving in the absence of a gravitationalanzvh13vam t.6 q:d0p field wit vyjk 9h;q8p2mw)1 bwch constant speed. It starts to accelerate by burning fuel. At the start, the fuel contributes to most of the mass hk)y8m2wb; pcqv91wjof the rocket. The burned fuel is ejected backwards from the rocket with a constant speed of v relative to the rocket and a constant rate of σ $kgs^{-1}$
. Which graph shows the variation of acceleration a of the rocket with time t as most of the fuel is used?

A. graph A
B. graph B
C. graph C
D. graph D


参考答案:  D


本题详细解析:

$\underline{\text{Explanation}}$:

The resultant force $F_{net}$ acting on the space rocket:

• $F_{net}=ma$

Where $m$ is the mass of the rocket at any given time.

When fuel is ejected from the rocket, it experiences a change in momentum in the frame of reference of the rocket.

A given mass $m$ of fuel has its velocity change from 0 relative to the rocket to its ejected speed of $v$.

Therefore the change of momentum of this mass of fuel is $p_2-p_1=mv-0=mv$

The force on the fuel causing this change in momentum is given by the equation

$F_{net}={\displaystyle \frac{\mathrm{\Delta }p}{\mathrm{\Delta }t}}={\displaystyle \frac{mv}{\mathrm{\Delta }t}}$

And we know that the rate of fuel ejected is $\sigma$ which equal to ${\displaystyle \frac{m}{\mathrm{\Delta }t}}$

Therefore the net force on the ejected fuel is

• $F_{net}=v\sigma$

Therefore

• $a={\displaystyle \frac{F_{net}}{M}}={\displaystyle \frac{v\sigma }{M}}$

where $M$ is the total initial mass of the rocket and $v$ is the velocity of the ejected fuel relative to the rocket.

The mass $m_f$ of the burned fuel after the time $t$:

• $m_f=\sigma t$

The total mass of the rocket decreases over time due to burned fuel ejection, the acceleration of the rocket after the time $t$:

    a=$\frac{v\sigma}{M-{\sigma}T}$

    According to this relationship, acceleration increases at an increasing rate. So the answer is D.

Qualitatively, the total mass of the rocket is constantly decreasing due to the ejected fuel. As the fuel is ejected at a constant rate, there is a constant net force on the rocket. With a constant net force and decreasing mass, there will be an increasing acceleration. At the start, the change in mass will have little effect on the acceleration, but over time the effect of the decrease in mass will become more and more significant and result in greater and greater acceleration.

• $a={\displaystyle \frac{v\sigma }{(M-\sigma t)}}$

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