A deorbit burn has been performed. During this deorbit burn a pre-calculated Delta V (change in velocity) of 290 ft/s (or 88.4 m/s) will be used to decrease the Shuttle’s altitude from 205 miles to 60 miles at perigee. The Shuttle’s Orbital Maneuvering System (OMS) engines provide a combined thrust of 12,000 force-pounds or 53,000 Newtons. The Shuttle weighs 2.50 x 105 lbs when fully loaded. The Shuttle has a mass of 1.13 x 105 kg when fully loaded.
Calculate how long a de-orbit burn must last in minutes and seconds to achieve the Shuttle’s change in altitude from 205 miles to 60 miles at perigee.
Method 1
Step 1: Solve for acceleration
From F=ma, rearrange to a=F/m
a=F/m= (53000 N)/(1.13 x 〖10〗^5 kg)= .469026549 〖m/s〗^2
Step 2: Solve for time
t=∆v/a=(88.4 m/s)/(.469026549 〖m/s〗^2 )=188.4754716 seconds
Step 3: Convert seconds to minutes
(188.4754716 seconds)/60=3.141257859 minutes
Step 4: Convert minutes to seconds
.141257859 × 60=8.47547154 seconds
Final answer:
It would require 3 minutes and 8 seconds of burn time to change the Shuttle’s altitude from 205 miles to 60 miles at perigee.
To calculate the duration of the deorbit burn, we need to use the rocket equation, which relates the change in velocity (ΔV) to the specific impulse (Isp), the mass of the spacecraft (m), and the propellant mass (mp). The equation is:
ΔV = Isp * g0 * ln(m0/mf),
Where:
- ΔV is the change in velocity (in meters per second)
- Isp is the specific impulse (in seconds)
- g0 is the standard acceleration due to gravity (9.81 m/s^2)
- m0 is the initial mass of the spacecraft (including propellant)
- mf is the final mass of the spacecraft (after propellant depletion)
In this case, the initial mass (m0) is the sum of the fully loaded mass of the Shuttle (1.13 x 10^5 kg) and the propellant mass (mp), and the final mass (mf) is the fully loaded mass without the propellant.
To find the propellant mass, we can take the ratio of the thrust force to the specific impulse:
mp = F * Δt / Isp,
Where:
- F is the thrust force (53,000 N)
- Δt is the duration of the burn we need to determine
Now we can rearrange the rocket equation to solve for the burn duration:
Δt = mp * Isp / F.
Let's proceed with the calculation.
First, convert the given ΔV from ft/s to m/s:
290 ft/s * 0.3048 m/ft = 88.392 m/s.
Next, calculate the propellant mass using the given thrust force and specific impulse:
mp = (53,000 N) * Δt / (Isp).
Since Δt is given in minutes and seconds, we need to convert it to seconds for the calculation. Let's assume the burn duration is t minutes and s seconds. So, Δt can be expressed as (60 * t + s) seconds.
Substituting this and the given values into the equation, we have:
88.392 m/s = (53,000 N) * (60 * t + s) / (Isp).
Now, solve for t and s. Rearranging the equation gives us:
t = (88.392 * Isp) / (53,000 * 60) - s / 60.
Let's calculate the burn duration in minutes and seconds.