a space-satellite mass of 600kg fires a rocket engine for 2 seconds expelling 40kg of exhaust gases at 350m/s. By how much will the speed of the capsule change? You may consider the initial relative velocities to be zero, but would it matter?
To determine the change in speed of the space satellite (capsule), we can use the principle of conservation of momentum. By firing the rocket engine, the expelled gases create a thrust force, causing the capsule to accelerate.
The momentum of an object is the product of its mass and velocity. Initially, the momentum of the capsule is zero since its relative velocity is considered to be zero. After firing the rocket engine, the expelled gases have a momentum equal to the mass of the exhaust gases times its velocity.
First, let's calculate the initial momentum of the exhaust gases:
Momentum of exhaust gases = mass of exhaust gases * velocity of exhaust gases
= 40 kg * 350 m/s
= 14,000 kg m/s
According to the principle of conservation of momentum, the change in momentum of the capsule will be equal in magnitude but opposite in direction to the change in momentum of the exhaust gases.
Change in momentum of capsule = -Change in momentum of exhaust gases
= -14,000 kg m/s
Since the change in momentum is equal to the force applied (due to Newton's second law of motion) multiplied by the time for which the force was applied, we can calculate the force applied by the expelled gases:
Force applied = Change in momentum / Time
= (-14,000 kg m/s) / 2 s
= -7,000 N
The negative sign indicates that the force is in the opposite direction of the exhaust gases expelled.
Now, to calculate the change in speed of the capsule, we can use the equation for the acceleration of an object under the influence of a constant force:
Acceleration = Force / Mass
In this case, the mass refers to the mass of the capsule. Therefore, the acceleration of the capsule can be calculated as:
Acceleration = -7,000 N / 600 kg
= -11.67 m/s^2
The negative sign indicates a decrease in speed.
Finally, we can calculate the change in speed of the capsule using the equation for uniformly accelerated motion:
Change in speed = Acceleration * Time
Change in speed = (-11.67 m/s^2) * 2 s
= -23.34 m/s
Therefore, the speed of the capsule will decrease by 23.34 m/s.
Now, let's address the question of whether the initial relative velocities matter. In this problem, the initial relative velocities are considered to be zero. However, even if the initial relative velocities were not zero, it would not affect the magnitude of the speed change of the capsule. The change in momentum of the capsule depends only on the magnitude and direction of the momentum of the expelled gases, which is determined by the mass and velocity of the gases being expelled. The initial relative velocities would only affect the overall velocities of the capsule and the exhaust gases, but not the change in velocity of the capsule.