A rocket of mass 50 kg is launched vertically. Its fuel is being burnt at a rate of 2 kg s-1 and its exhaust gas is being forced out with a speed of 1000 m s-1. What is the initial acceleration of the rocket?
mv rocket = m v exhaust
so
d/dt (mv rocket) = d/dt (mv exhaust)
m v rocket = (50 - 2 t)v = m v exhaust = (2t)1000
50 v - 2vt = 2000 t
50 dv/dt -2v - 2t dv/dt = 2000
at t = 0 and v = 0
50 dv/dt = 50 a = 2000
a = 40 m/s^2
To find the initial acceleration of the rocket, we need to apply Newton's second law of motion, which states that the force applied on an object is equal to its mass multiplied by its acceleration.
Given information:
Mass of the rocket (m) = 50 kg
Rate of fuel burnt (dm/dt) = 2 kg/s
Speed of exhaust gas (v) = 1000 m/s
The force acting on the rocket can be calculated by multiplying the mass of the exhaust gas expelled per second (dm/dt) by the velocity of the exhaust gas (v).
Force (F) = (dm/dt) * v
Substituting the given values:
F = 2 kg/s * 1000 m/s
F = 2000 N
Since there are no other external forces acting on the rocket, the force exerted by the exhaust gas is the only force propelling the rocket. Therefore, this force (F) is equal to the product of the mass of the rocket (m) and its acceleration (a).
F = m * a
Substituting the values:
2000 N = 50 kg * a
To find the acceleration (a), we can rearrange the equation:
a = 2000 N / 50 kg
a = 40 m/s^2
Therefore, the initial acceleration of the rocket is 40 m/s^2.