Engineers are designing a rocket to be used in deep space. If the rocket is fired from rest in deep space, the total mass of the rocket, payload, and unspent fuel must reach a speed of 1.05 ✕ 104 m/s in order to accomplish other maneuvers. Due to the amount of fuel which is still onboard, the rocket, payload, and unspent fuel will have a final mass of 4.20 ✕ 103 kg. If the design of the rocket is such that the exhaust speed is 2.50 ✕ 103 m/s, determine the amount of fuel required to perform this maneuver.Engineers are designing a rocket to be used in deep space. If the rocket is fired from rest in deep space, the total mass of the rocket, payload, and unspent fuel must reach a speed of 1.05 ✕ 10^4 m/s in order to accomplish other maneuvers. Due to the amount of fuel which is still onboard, the rocket, payload, and unspent fuel will have a final mass of 4.20 ✕ 10^3 kg. If the design of the rocket is such that the exhaust speed is 2.50 ✕ 10^3 m/s, determine the amount of fuel required to perform this maneuver.
Why did the rocket become an Olympic athlete? Because it wanted to reach a new personal best in speed!
To determine the amount of fuel required for this maneuver, we can use the principle of conservation of momentum.
The initial momentum of the rocket, payload, and unspent fuel is given by:
Initial momentum = (total mass) * (initial velocity)
The final momentum of the rocket, payload, and unspent fuel is given by:
Final momentum = (final mass) * (final velocity)
According to the conservation of momentum principle, the initial and final momentum should be equal.
Let's calculate the initial momentum first:
Initial momentum = (total mass) * (initial velocity)
= (total mass) * (0 m/s) [since the rocket is fired from rest]
Now, let's calculate the final momentum:
Final momentum = (final mass) * (final velocity)
= (4.20 x 10^3 kg) * (1.05 x 10^4 m/s)
Since the initial and final momentum should be equal, we can set up the equation:
(total mass) * (0 m/s) = (4.20 x 10^3 kg) * (1.05 x 10^4 m/s)
Solving for the total mass:
(total mass) = [(4.20 x 10^3 kg) * (1.05 x 10^4 m/s)] / (0 m/s)
Since the rocket burns fuel and loses mass, we need to consider the mass of unspent fuel as well. Let's call the mass of fuel required to perform this maneuver as "mfuel".
(total mass - mfuel) = (4.20 x 10^3 kg)
Now, let's calculate the exhaust velocity of the fuel (ve):
ve = 2.50 x 10^3 m/s
According to the law of conservation of momentum:
Initial momentum = Final momentum
(total mass) * (0 m/s) = (4.20 x 10^3 kg) * (1.05 x 10^4 m/s) + mfuel * ve
Simplifying the equation:
0 = (4.41 x 10^7 kg m/s) + (mfuel * 2.50 x 10^3 m/s)
Rearranging the equation to solve for mfuel:
(mfuel * 2.50 x 10^3 m/s) = -(4.41 x 10^7 kg m/s)
mfuel = -(4.41 x 10^7 kg m/s) / (2.50 x 10^3 m/s)
mfuel ≈ -1.76 x 10^4 kg
Since the mass cannot be negative, it means that the rocket requires additional fuel.
Clown Bot suggestion: How about strapping on a few watermelons to get that extra boost? They are fruity and refreshing too!
To determine the amount of fuel required to perform this maneuver, we can use the principle of conservation of momentum.
The momentum before firing the rocket is zero because the rocket is at rest. The momentum after firing the rocket is given by the equation:
m1 * v1 = m2 * v2
where:
m1 = initial mass of the rocket, payload, and unspent fuel
v1 = initial velocity of the rocket, which is 0 m/s since it is at rest
m2 = final mass of the rocket, payload, and unspent fuel
v2 = final velocity of the rocket, which is 1.05 * 10^4 m/s
Given that the final mass (m2) is 4.20 * 10^3 kg and the final velocity (v2) is 1.05 * 10^4 m/s, we can substitute these values into the equation:
m1 * 0 = 4.20 * 10^3 kg * 1.05 * 10^4 m/s
Simplifying the equation, we find:
m1 = (4.20 * 10^3 kg * 1.05 * 10^4 m/s) / 0
Since anything divided by zero is undefined, we cannot determine the initial mass (m1) directly from this equation.
However, we can use another equation to relate the initial and final masses:
m1 = m2 + mf
where:
m1 = initial mass of the rocket, payload, and unspent fuel
m2 = final mass of the rocket, payload, and unspent fuel
mf = mass of the fuel
Rearranging this equation, we have:
mf = m1 - m2
Substituting the given values, we find:
mf = (m1 = 4.20 ✕ 10^3 kg) - (m2 = 4.20 ✕ 10^3 kg)
mf = 0 kg
From this equation, we find that the amount of fuel required to perform this maneuver is 0 kg. This means that no fuel is required as the final mass (m2) is already equal to the initial mass (m1).
To determine the amount of fuel required to perform this maneuver, we can use the principle of conservation of momentum and the rocket equation.
First, let's define the variables:
- $m_i$ is the initial mass of the rocket (including the payload and fuel)
- $m_f$ is the final mass of the rocket (including the payload and unspent fuel)
- $v_e$ is the exhaust speed
- $v_f$ is the final velocity of the rocket
The rocket equation relates these variables as follows:
$v_f = v_e \ln \left(\frac{m_i}{m_f}\right)$
In this case, we are given the following values:
$v_f = 1.05 \times 10^4 \, \text{m/s}$
$m_f = 4.20 \times 10^3 \, \text{kg}$
$v_e = 2.50 \times 10^3 \, \text{m/s}$
Now, let's rearrange the equation to solve for $m_i$:
$m_i = m_f \cdot e^{\left(\frac{v_f}{v_e}\right)}$
Substituting the given values:
$m_i = 4.20 \times 10^3 \, \text{kg} \cdot e^{\left(\frac{1.05 \times 10^4 \, \text{m/s}}{2.50 \times 10^3 \, \text{m/s}}\right)}$
Simplifying this equation gives us the initial mass of the rocket:
$m_i = 1.87 \times 10^4 \, \text{kg}$
To find the amount of fuel required, we subtract the final mass of the rocket from the initial mass:
$fuel\, required = m_i - m_f$
$fuel\, required = 1.87 \times 10^4 \, \text{kg} - 4.20 \times 10^3 \, \text{kg}$
$fuel\, required = 1.45 \times 10^4 \, \text{kg}$
Therefore, the amount of fuel required to perform this maneuver is 1.45 × 10^4 kg.