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

note: the given question is incomplete.

complete question is given below.

which is an example of kinetic energy?

a. the energy stored in ethanol  

b. a hockey puck sliding across the ice  

c. a compressed spring  

d. a ball sitting at the top of a ramp

solution: the kinetic energy is the energy posses by the virtue of the motion of the object.

in the given options, a, c and d are the example of potential energy because it is the energy posses by the virtue of the rest of the object. but option b has the kinetic energy as the object is in motion.

hence, a hockey puck sliding across the ice   is the apex example of kinetic energy.

so, the option (b) a hockey puck sliding across the ice   is the correct option.

get the total number of pieces of fruit and divide it by 4 to get the average. the result is the answer to the problem. 14 total number of fruits and there are 4 members. 14 divided by 4 is equals to 3.5.