Question

ana gonzales
to: student physicists
re: pod and space station data
youve done excellent work in investigating how collisions affect different objects, and were looking forward to your final report.
in case you might find the following helpful for comparing the speed of the pod and the space station after the collision, here are their masses;
pod: 1,100 kg space station: 415,400 kg
regards,
ana
dr. ana gonzales, program scientist
asteroid collection mission
how can you use this information about mass to help you compare the speed of the pod after the collision to the speed of the space station after the collision?

Explanation:

Step1: Recall the law of conservation of momentum

The law of conservation of momentum states that \(m_1v_1 + m_2v_2=m_1v_1'+m_2v_2'\). Before the collision, assume the space - station is at rest (\(v_2 = 0\)), and let the initial velocity of the pod be \(v_1\). After the collision, the velocities are \(v_1'\) (pod) and \(v_2'\) (space - station). So \(m_1v_1=m_1v_1'+m_2v_2'\).

Step2: Analyze the relationship between mass and velocity after collision

We can rewrite the equation as \(m_1(v_1 - v_1')=m_2v_2'\). Since \(m_1 = 1100\space kg\) (pod) and \(m_2=415400\space kg\) (space - station) and \(m_2\gg m_1\). From the equation \(v_2'=\frac{m_1}{m_2}(v_1 - v_1')\). Because \(m_2\) is much larger than \(m_1\), the velocity of the space - station \(v_2'\) after the collision will be much smaller than the change in velocity of the pod (\(\vert v_1 - v_1'\vert\)). In a collision, the impulse (change in momentum) on both objects is the same in magnitude (\(F\Delta t=\Delta p\)). Using \(\Delta p = m\Delta v\), for the same \(\Delta p\), \(m\) and \(\Delta v\) are inversely related.

Answer:

Since the impulse (change in momentum) on the pod and the space - station during the collision is the same in magnitude (from Newton's third law, the force exerted by the pod on the space - station is equal in magnitude and opposite in direction to the force exerted by the space - station on the pod, and they act for the same time interval). Using the formula \(\Delta p=m\Delta v\) (where \(\Delta p\) is the change in momentum, \(m\) is mass and \(\Delta v\) is the change in velocity), for the same \(\Delta p\), mass \(m\) and change in velocity \(\Delta v\) are inversely related. Given that the mass of the space - station (\(m = 415400\space kg\)) is much larger than the mass of the pod (\(m = 1100\space kg\)), the change in velocity (and thus the final velocity, assuming one was initially at rest) of the space - station will be much smaller than that of the pod. So the speed of the pod after the collision will be much larger than the speed of the space - station after the collision.