Answer :
The initial mass (mi) is 75000 kg and the initial velocity (vi) is approximately -8074.9 m/s.
To solve this problem, we can use the concept of the rocket equation. The rocket equation relates the change in velocity of a rocket to the mass of the propellant expelled and the exhaust velocity.
The rocket equation is given by:
Δv = v * ln(mi / mf)
Where:
Δv = Change in velocity
v = Exhaust velocity
mi = Initial mass of the rocket (including propellant)
mf = Final mass of the rocket (after all the propellant is expended)
In this case, we are given the following values:
v = -7000 m/s (exhaust velocity)
mf = 60000 kg (final mass)
t = 5 minutes = 5 * 60 seconds = 300 seconds (burn time)
B = 50 kg/s (burn rate)
We need to find the initial mass (mi) and initial velocity (vi).
Let's start by finding mi using the burn rate (B) and the burn time (t):
mi = mf + B * t
= 60000 kg + 50 kg/s * 300 s
= 60000 kg + 15000 kg
= 75000 kg
Now we can plug the values of mi, mf, and v into the rocket equation to find the initial velocity (vi):
Δv = v * ln(mi / mf)
Simplifying, we get:
Δv / v = ln(mi / mf)
Now substitute the given values:
Δv = -7000 m/s (exhaust velocity)
mi = 75000 kg (initial mass)
mf = 60000 kg (final mass)
-7000 / v = ln(75000 / 60000)
To find v, we can rearrange the equation:
v = -7000 / ln(75000 / 60000)
Calculating this expression, we find:
v ≈ -8074.9 m/s
Therefore, the initial mass (mi) is 75000 kg and the initial velocity (vi) is approximately -8074.9 m/s.
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