A 1000-kg space probe is motionless in space. To start moving, its main engine is fired for 5 seconds during which time it ejects exhaust gases at 5000 m/s. At the end of this process it is moving at 20 m/s. The approx. mass of the ejected gas is??
To solve this problem, we can use the laws of conservation of momentum.
The initial momentum of the system (space probe + ejected gas) is zero since the space probe is motionless. The final momentum is the sum of the momentum of the space probe and the ejected gas.
Given:
Initial velocity of the space probe (u1) = 0 m/s
Final velocity of the space probe (v1) = 20 m/s
Velocity of the ejected gas (v2) = 5000 m/s
The mass of the space probe (m1) = 1000 kg
The mass of the ejected gas (m2) = ?
The conservation of momentum can be expressed as:
(m1 * u1) + (m2 * u2) = (m1 * v1) + (m2 * v2)
Substituting the given values:
(1000 kg * 0 m/s) + (m2 * 5000 m/s) = (1000 kg * 20 m/s) + (m2 * 5000 m/s)
0 + (5000 m/s * m2) = (1000 kg * 20 m/s) + (5000 m/s * m2)
Simplifying the equation:
5000 m/s * m2 = 20000 kg*m/s + 5000 m/s * m2
Rearranging the terms:
5000 m/s * m2 - 5000 m/s * m2 = 20000 kg*m/s
0 = 20000 kg*m/s
Since this is not possible, there seems to be an error in the problem statement or the given values. Please double-check the information provided and let me know if you have any additional details.
To find the approximate mass of the ejected gas, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of a system remains constant unless acted upon by external forces. In this case, the system consists of the space probe and the ejected gas.
The momentum of an object can be calculated by multiplying its mass and velocity.
Before the firing of the engine, the space probe is motionless. Therefore, its momentum is zero (mass * velocity = 0).
After the firing of the engine, the space probe is moving at 20 m/s. The momentum of the space probe can be calculated as follows:
Momentum of space probe = mass of space probe * velocity of space probe
Momentum of space probe = 1000 kg * 20 m/s = 20,000 kg·m/s
According to the conservation of momentum, the total momentum of the system before and after the ejection of gas must be equal.
Therefore, the momentum of the ejected gas can be calculated as follows:
Momentum of ejected gas = momentum of the space probe
Momentum of ejected gas = 20,000 kg·m/s
Now, let's calculate the mass of the ejected gas.
Momentum of ejected gas = mass of ejected gas * velocity of ejected gas
20,000 kg·m/s = mass of ejected gas * 5000 m/s
Rearranging the equation, we get:
mass of ejected gas = 20,000 kg·m/s / 5000 m/s
mass of ejected gas = 4 kg
Therefore, the approximate mass of the ejected gas is 4 kg.