a rocket of mass 1000kg containing a propellant gas of 3000kg is to be launched vertically. If the fuel is consumed at a steady rate of 60kg per seconds, calculate the least velocity of the exhaust gas if the rocket and content will just lift off the launching pad immediately after firing?
Ft=M(V-U). Divide all by T. F=M\T(∆V). Mass one=1000kg Mass two =3000kg. Total mass =1000+3000=4000kg. Acceleration =10m\s^2.Force =M×A=4000×10=40000. Rate=M\T=60kg\s. 40000=60(∆V). Velocity =666•67m\s.
F = d momentum/dt = mass/second * v
so
F = 60 kg/s * v
= m g = (1000+3000) (9.81)
so
v = 4000*9.81 / 60
1000AND3000
force=thrust=mg
massrate*v=mg
60kg/sec*v=4000kg*9.8N/kg
solve for v
To calculate the least velocity of the exhaust gas required for the rocket to lift off, we can use the concept of momentum.
Let's denote:
m1 = mass of the rocket (1000 kg)
m2 = mass of the propellant gas (3000 kg)
m3 = mass of the exhaust gas expelled per second (60 kg/s)
Initially, the total mass of the system (rocket + propellant) is:
m_total = m1 + m2 = 1000 kg + 3000 kg = 4000 kg
As the propellant is consumed, the total mass decreases at a rate of m3 kg/s. After t seconds, the mass of the system will be:
m_total = 4000 kg - m3t
According to the law of conservation of momentum, the initial momentum of the system must be equal to the momentum of the exhaust gas expelled per second.
Initial momentum of the system:
p_initial = m_total * v_initial
Where v_initial is the initial velocity of the system (rocket + propellant) with respect to the ground.
Momentum of the exhaust gas expelled per second:
p_exhaust = m3 * v_exhaust
Since there is no external force acting on the system in the vertical direction (assuming no air resistance), the change in momentum of the system (p_initial - p_exhaust) will determine the acceleration needed to lift off.
Change in momentum:
p_initial - p_exhaust = (m_total * v_initial) - (m3 * v_exhaust)
To lift off, the change in momentum must be greater than or equal to zero. So we can set up the equation:
(m_total * v_initial) - (m3 * v_exhaust) ≥ 0
Substituting the values we have:
(4000 kg - m3t) * v_initial ≥ m3 * v_exhaust
Now, we need to find the minimum velocity of the exhaust gas that allows the rocket to lift off immediately after firing. This means that t should be equal to 0 (no time elapses).
Therefore, the equation becomes:
(4000 kg) * v_initial ≥ m3 * v_exhaust
Now we can substitute the given values:
4000 kg * v_initial ≥ 60 kg/s * v_exhaust
Dividing both sides of the equation by 4000 kg:
v_initial ≥ (60 kg/s * v_exhaust) / 4000 kg
Simplifying:
v_initial ≥ (3/200) * v_exhaust
So, the least velocity of the exhaust gas, v_exhaust, required for the rocket to lift off is equal to or greater than v_initial times (3/200).
A rocket of mass 1000kg contaning a propellant gas of 3000kg
F=d/dt
Mg=d/dt
4000*10=60v
v=666.7