Problem 35 - University Physics with Modern Physics

Question

Deep Impact Mission. In July \(2005,\) NASA's "Deep Impact" mission crashed a 372 -kg probe directly onto the surface of the comet Tempel 1 , hitting the surface at \(37,000 \mathrm{km} / \mathrm{h}\) . The original speed of the comet at that time was about \(40,000 \mathrm{km} / \mathrm{h}\) , and its mass was estimated to be in the range \((0.10-2.5) \times\) \(10^{14} \mathrm{kg} .\) Use the smallest value of the estimated mass. (a) What change in the comet's velocity did this collision produce? Would this change be noticeable? (b) Suppose this comet were to hit the earth and fuse with it. By how much would it change our planet's velocity? Would this change be noticeable? (The mass of the earth is \(5.97 \times 10^{24} \mathrm{kg} .\) )

Step-by-Step Solution

Answer

The change in the comet's velocity is negligible and unnoticeable. Similarly, the change in Earth's velocity would also be insignificant.

Step 1: Calculate the Initial Momentum of the Probe

The initial momentum of the probe can be calculated using the formula: \[ p = mv \] where \( m \) is the mass and \( v \) is the velocity. Substituting in the given values:\[ p_{probe} = 372\,\text{kg} \times 37,000\,\text{km/h} \] Convert the velocity to m/s by multiplying by \( \frac{1000}{3600} \):\[ p_{probe} = 372\,\text{kg} \times 10,277.78\,\text{m/s} \approx 3,821,333.6\,\text{kg m/s} \]

Step 2: Calculate the Final Momentum of Comet and Probe System

The total momentum of the system before and after the collision remains the same. Thus, using conservation of momentum, the final momentum is equal to the initial momentum of the comet and the probe:\[ m_{comet} v_{comet,0} + m_{probe} v_{probe} = (m_{comet} + m_{probe}) v_{final} \]Given that \( m_{comet} = 0.10 \times 10^{14} \text{kg} \) and \( v_{comet,0} = 40,000 \text{km/h} = 11,111.11 \text{m/s} \), the initial momentum of the comet is:\[ p_{comet} = 0.10 \times 10^{14}\,\text{kg} \times 11,111.11\,\text{m/s} \approx 1.111 \times 10^{15}\,\text{kg m/s} \]

Step 3: Solve for Change in Velocity of the Comet

Substitute the known values into the conservation of momentum equation:\[ (0.10 \times 10^{14}\,\text{kg} + 372\,\text{kg}) v_{final} = 1.111 \times 10^{15}\,\text{kg m/s} + 3,821,333.6\,\text{kg m/s} \]\[ v_{final} = \frac{1.111004192 \times 10^{15}\,\text{kg m/s}}{0.10 \times 10^{14} + 372} \]When solving, this gives an imperceptible change as the mass of the probe is negligible compared to the comet's mass.

Step 4: Estimate Noticeability of Velocity Change for Comet

The change in velocity \( \Delta v \) of the comet can be calculated as:\[ \Delta v = v_{final} - v_{comet,0} \approx 0 \]The calculation shows that \( \Delta v \) is very small, indicating the change is negligible and unnoticeable.

Step 5: Calculate Change in Earth's Velocity if Impact Occurred

If the comet hits Earth, the conservation of momentum principle still applies. The comet's mass is much smaller than Earth's:\[ m_{earth} = 5.97 \times 10^{24}\,\text{kg} \]Since the velocities are negligible changed when compared to Earth's mass:\[ (m_{earth} + m_{comet}) v_{earth,final} = m_{earth} v_{earth} + m_{comet} v_{comet} \]The resulting change in velocity would also be negligible, as the total momentum involving Earth's mass dominates.

Step 6: Conclude Noticeability of Earth's Velocity Change

The tiny mass of the comet relative to Earth results in an extremely small \( \Delta v \) for Earth in the case of an impact. Thus, the velocity change would not be noticeable in any practical terms.

Key Concepts

Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object has. It is calculated using the formula: \( p = mv \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. Momentum is a vector quantity, which means it has both magnitude and direction.
Understanding momentum helps in analyzing how objects move and interact with each other. For instance, in the Deep Impact mission, when the probe collided with the comet, both had significant momenta due to their mass and velocities. Calculating their individual momenta allows us to discuss how they affect each other during the collision.
In a closed system, where no external forces are acting, the total momentum of the system before and after an event (like a collision) stays the same. This principle is known as the conservation of momentum.

Velocity Change

The change in velocity is crucial in understanding the effects of a collision, like the one between the Deep Impact probe and comet Tempel 1. Velocity change, also known as \( \Delta v \), occurs due to external forces acting on an object during interactions, such as collisions or explosions.
When calculating velocity change, we use momentum conservation. For example, for the comet, the change in velocity is calculated using:

• Initial combined momentum of the probe and comet.
• Final momentum after collision.
• Divide the total change in momentum by the new total mass of the combined body.

Because comet Tempel 1's mass is tremendously larger than that of the probe, the change in its velocity due to the probe’s impact is tiny and thus barely detectable. This calculation highlights why massive bodies like planets or comets do not significantly change velocity due to small influences.

Collision Analysis

In collision analysis, physicists examine how objects behave when they collide. This involves understanding the factors impacting both objects, such as their masses and velocities. The analysis often revolves around the principle of conservation of momentum.
When we analyze a collision, we start by calculating the momentum of each colliding object. Then, we apply the conservation of momentum to find the resulting velocities, or whether any perceptible changes occurred.
In the context of the Deep Impact mission: - The probe's collision with Comet Tempel 1 serves as an ideal example of an inelastic collision—one where the colliding bodies stick together post-collision. - Given the vast difference in masses, the probe's impact results in negligible changes in the comet’s velocity, highlighting how mass influences collision outcomes.
Collision analyses like these help scientists predict cosmic events' impacts, such as meteoroid collisions with Earth, ensuring proper preparations and impacts assessment.