The solid fuel of a 300.kg rocket travelling at 1.64x104 km/h is ignited to correct the rocket trajectory in mid-flight to Mars. 10.00 kg of fuel is burnt in 15.00 s. If the exhaust velocity of the fuel, relative to the rocket, is 2.90x103 km/h, what is the acceleration of the rocket (in m/s2) at the start of this burn?
To find the acceleration of the rocket at the start of the burn, we can use the principle of conservation of momentum.
Let's break down the steps to get the answer:
Step 1: Convert the speed and velocity values to m/s:
- The rocket's speed is given as 1.64x10^4 km/h. We need to convert it to m/s by multiplying by (1000 m)/(1 km) and (1 h)/(3600 s).
1.64x10^4 km/h = (1.64x10^4 km/h) * (1000 m/km) * (1 h/3600 s) = 455.56 m/s
- The exhaust velocity of the fuel relative to the rocket is given as 2.9x10^3 km/h. Convert it to m/s using the same conversion factors above:
2.9x10^3 km/h = (2.9x10^3 km/h) * (1000 m/km) * (1 h/3600 s) = 805.56 m/s
Step 2: Calculate the change in velocity (delta V) of the rocket:
Delta V = exhaust velocity * natural logarithm (initial mass/final mass)
The initial mass is the mass of the rocket plus the mass of the fuel, which is 300 kg + 10 kg = 310 kg.
The final mass is the mass of the rocket after burning the fuel, which is 300 kg.
Therefore, Delta V = 805.56 m/s * ln(310/300) = 805.56 m/s * ln(1.03333) = 805.56 m/s * 0.03266 = 26.31 m/s
Step 3: Calculate the acceleration using the rocket equation:
Acceleration = Delta V / burn time
The burn time is given as 15.00 s.
Acceleration = 26.31 m/s / 15.00 s = 1.754 m/s^2
Therefore, the acceleration of the rocket at the start of the burn is 1.754 m/s^2.
To find the acceleration of the rocket at the start of the burn, we can use the conservation of momentum principle.
The momentum of the rocket before the burn is equal to the momentum of the rocket plus the momentum of the burnt fuel after the burn. Mathematically, this can be expressed as:
mrocket * vrocket = (mrocket + mfuel) * vfinal
where:
mrocket = mass of the rocket = 300 kg
vrocket = velocity of the rocket = 1.64x10^4 km/h = 4.56x10^3 m/s (converted to m/s)
mfuel = mass of the burnt fuel = 10.00 kg
vfinal = velocity of the rocket plus the burnt fuel after the burn
Now, we need to calculate vfinal by considering the velocity of the burnt fuel relative to the rocket:
vfinal = vrocket + vexhaust
where:
vexhaust = exhaust velocity of the fuel relative to the rocket = 2.90x10^3 km/h = 8.06x10^2 m/s (converted to m/s)
Substituting the known values into the equation for vfinal:
vfinal = 4.56x10^3 m/s + 8.06x10^2 m/s = 5.37x10^3 m/s
Now we can substitute the values of mrocket, vrocket, mfuel, and vfinal into the momentum equation to solve for the acceleration:
(300 kg) * (4.56x10^3 m/s) = (300 kg + 10.00 kg) * (5.37x10^3 m/s)
137.89x10^3 kg*m/s = 310.00x10^3 kg*m/s
Dividing both sides of the equation by the total mass of the rocket and the burnt fuel:
137.89x10^3 kg*m/s / 310.00x10^3 kg = a
a ≈ 0.445 m/s^2
Therefore, the acceleration of the rocket at the start of the burn is approximately 0.445 m/s^2.