# Find the terminal speed of a 750-kg rocket

In summary, the formula used to find the terminal speed of a rocket is v_f = v_i + v_{ex}*ln{\frac{M_i}{M_f}}, where v_f is the final velocity, v_i is the initial velocity (which is zero if the rocket starts from rest), v_ex is the relative velocity of the ejected fuel, and M_i and M_f represent the initial and final mass of the rocket. Using this formula, the terminal speed of a 750-kg rocket that starts from rest carrying 2600kg of fuel and expels its exhaust gases at 1.8 k/ma is calculated to be 2.2 km/s. It is important to note that the initial mass includes the mass
Find the terminal speed of a 750-kg rocket that starts from rest carrying 2600kg of fuel and that expels its exhuast gases at 1.8 k/ma

here's the formula I am using: $$v_f = v_i + v_{ex}*ln{\frac{M_i}{M_f}}$$

well since it's at rest, initial velocity is zero. so...

$$v_f = 1.8*ln(750/2500)$$ which turns out to be 2.2 km/s. but the book says the answer is 2.7km/s

I believe the Mi and Mf mean mass initial and mass final, check those, the initial mass will be the mass of the rocket and its fuel and the final mass will be just the rocket...I think Guess I'm a little tired again today :zzz:

michael376071 said:
I believe the Mi and Mf mean mass initial and mass final, check those, the initial mass will be the mass of the rocket and its fuel and the final mass will be just the rocket...I think Guess I'm a little tired again today :zzz:

Yah that is correct. Mi is mass initial (overall mass) and Mf is mass final (overall mass - expelled gas)

If I denote the relative velocity of the ejected fuel with respect to the rocket as $\u$ then the variable mass equation I use is

$$M\frac{dv}{dt} = F_{ext}-u\frac{dM}{dt}$$

In case of a rocket in the gravitational field (I guess this is what you want?), $F_{ext} = Mg$ so here,

$$M\frac{dv}{dt} = Mg-u\frac{dM}{dt}$$

At terminal speed, the acceleration of the rocket is zero...

## 1. What is terminal speed?

Terminal speed, also known as terminal velocity, is the maximum speed that an object can reach when falling through a fluid, such as air or water. It occurs when the drag force of the fluid is equal to the force of gravity acting on the object.

## 2. How is terminal speed calculated?

The formula for calculating terminal speed is v = √(2mg/ρACd), where v is the terminal speed, m is the mass of the object, g is the acceleration due to gravity, ρ is the density of the fluid, A is the cross-sectional area of the object, and Cd is the drag coefficient.

## 3. What factors affect the terminal speed of a rocket?

The terminal speed of a rocket is affected by several factors, including the mass and shape of the rocket, the density and viscosity of the fluid it is falling through, and the force of gravity. Additionally, any external forces acting on the rocket, such as wind or thrust, can also impact its terminal speed.

## 4. How can the terminal speed of a 750-kg rocket be determined?

To determine the terminal speed of a 750-kg rocket, you will need to know the values for the variables in the terminal speed formula and plug them in to calculate the speed. For example, if the rocket has a cross-sectional area of 5 square meters, a drag coefficient of 0.5, and is falling through air with a density of 1.2 kg/m3, the terminal speed would be approximately 108 m/s.

## 5. Can the terminal speed of a rocket ever be exceeded?

No, the terminal speed is the maximum speed that an object can reach when falling through a fluid. If a rocket were to continue to accelerate beyond its terminal speed, the drag force would increase and eventually become equal to the force of gravity, causing the rocket to maintain a constant speed.

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