A rocket expels a gas at rate of 0.5kg/s if the force produced by the rocket is 2000N what is the velocity with which gas expelled
F = d/dt momentum = d/dt (m v) = v dm/dt + m dv/dt
v is constant
so
F = v dm/dt =
2000 = v (0.5)
v = 4000 m/s
To determine the velocity with which the gas is expelled from the rocket, we can use the principle of conservation of momentum. The momentum of the gas before it is expelled is equal to the momentum of the rocket after the gas is expelled.
Given:
Mass of gas expelled per second = 0.5 kg/s
Force produced by the rocket = 2000 N
Step 1: Calculate the change in momentum caused by the gas expelled per second.
Change in momentum = Mass x velocity of the gas
Change in momentum = (0.5 kg/s) x (velocity of the gas)
Step 2: Calculate the change in momentum caused by the rocket.
Change in momentum = (mass of the rocket) x (change in velocity of the rocket)
As the mass of the rocket is constant, the change in velocity of the rocket is equal to the velocity of the gas expelled.
Step 3: Equate the change in momentum caused by the gas and the change in momentum caused by the rocket.
Change in momentum caused by the gas = Change in momentum caused by the rocket
(0.5 kg/s) x (velocity of the gas) = (mass of the rocket) x (velocity of the gas)
Step 4: Solve for the velocity of the gas.
0.5 kg/s x velocity of the gas = (mass of the rocket) x (velocity of the gas)
0.5 kg/s = mass of the rocket
velocity of the gas = 2000 N / 0.5 kg/s
Calculating the velocity of the gas expelled:
velocity of the gas = 2000 N / 0.5 kg/s = 4000 m/s
Therefore, the velocity with which the gas is expelled from the rocket is 4000 m/s.
To find the velocity with which gas is expelled from the rocket, we can use the equation of motion known as Newton's second law of motion. According to this law, the force produced by an object is directly proportional to the rate at which momentum changes.
The equation for momentum is given by the formula:
Momentum = mass x velocity
In this case, the mass refers to the mass of the gas being expelled from the rocket, and the velocity is the velocity of the expelled gas. We can rearrange the formula to solve for velocity:
Velocity = Momentum / mass
In our case, the rate at which gas is expelled from the rocket is given as 0.5 kg/s. So, in one second, the mass of the expelled gas would be 0.5 kg. The momentum of the expelled gas can be calculated using the force produced by the rocket. Since force is the change in momentum per unit time, we can rearrange the formula:
Force = Change in Momentum / Time
Rearranging further:
Change in Momentum = Force x Time
Substituting the given values, with the force as 2000 N and time as 1 second:
Change in Momentum = 2000 N x 1 s
Now, we have the change in momentum, which we can use to find the velocity:
Velocity = Change in Momentum / mass
Velocity = (2000 N x 1 s) / 0.5 kg
Simplifying:
Velocity = 4000 Ns / 0.5 kg
The unit "Ns" is equivalent to "kg m/s" (Newton-second is the unit of momentum), so we can rewrite the equation:
Velocity = 4000 kg m/s / 0.5 kg
Canceling out the units of "kg" and performing the division:
Velocity = 4000 m/s
Therefore, the velocity with which the gas is expelled from the rocket is 4000 m/s.