Consider a rocket (initial mass m o) accelerating from rest in free space. At first, as it speeds up, its momentum p increases, but as its mass m decreases p eventually begins to decrease. For what value of m is p maximum

m = m_o/e

Explanation:

The rocket equation is given as;

m(dv/dt) = -v_ex(dm/dt)

v_ex is the exhaust velocity

Now, formula for momentum is;

p = mv

Differentiating with respect to time, we have;

dp/dt = m'v + mv'

Where;

m' is mass rate

v' is rate of change in velocity

Since we are dealing with exhaust velocity and momentum(p) is Maximum when v = v_ex then mv' can also be written as: -m'v_ex.

Thus;

dp/dt = m'v - m'v_ex

dp/dt = m'(v - v_ex)

Now, from the rocket equation we dt will cancel out to give;

-dv/v_ex = dm/m

Integrating both sides;

-∫dv/v_ex = ∫dm/m

This gives;

-v/v_ex = In(m/m_o)

Since momentum(p) is Maximum when v = v_ex

Thus;

-v/v = In(m/m_o)

-1 = In(m/m_o)

e^(-1) = m/m_o

m = m_o(e^(-1))

m = m_o/e

i would say the answer is food.

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-first use of geothermal energy occurred more than 10,000 years ago in north america by american paleo-indians

-the first industrial use of geothermal energy began near pisa, italy in late 18th century

-in 1904, italian scientist piero ginori conti invented the first geothermal electric power plant in which steam was used to generate the power

-with the above experiment, the first geothermal plant in usa started in 1922 with a capacity of 250 kilowatts

-however, in 1946 first ground-source geothermal heat pump installed at commonwealth building in portland, oregon

-during the 1960’s, pacific gas and electric began operation of first large scale geothermal power plant in san francisco, producing 11 megawatts

-in 1973, when oil crisis began many countries began looking for renewable energy sources and by 1980’s geothermal heat pumps (ghp) started gaining popularity in order to reduce heating and cooling costs

-as effect of climate change started showing results, governments of various countries joined hands to fight against it, for which kyoto protocol was signed in japan in 1997, laid out emission targets for rich countries and required that they transfer funds and technology to developing countries, 184 countries have ratified it

hope this