Question

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 km/h. The original speed of the comet at that time was about 40,000 km/h, and its mass was estimated to be in the range (0.10 - 2.5) \(\times\)10\(^{14}\) 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\(^{24}\) kg.)

Short answer

The comet's velocity change is negligible due to its massive size. The Earth's velocity change would also be insignificantly small.

Step-by-step solution

Understand the Problem

We're answering parts (a) and (b) of the problem. For part (a), we need to calculate the change in the comet's velocity post collision using the smallest mass estimate. For part (b), we'll determine the change in Earth’s velocity if it collides with the comet.

Calculate Initial Values

The initial velocity of the comet, \( v_{comet} \), is 40,000 km/h and the mass of the comet, \( m_{comet} \), is estimated at 0.10 \( \times 10^{14} \) kg. The mass of the probe is 372 kg with an initial velocity, \( v_{probe} \), of 37,000 km/h.

Apply Conservation of Momentum for the Comet

The conservation of momentum formula is \( m_{comet} \times v_{comet} + m_{probe} \times v_{probe} = m_{comet} \times v_{comet,final} \). Use this to calculate the final velocity of the comet post-impact.

Solve for the Change in the Comet's Velocity

Rearrange the formula to find \( v_{comet,final} \) and compute the change, \( \Delta v_{comet} = v_{comet,final} - v_{comet} \). This gives an idea if the velocity change is noticeable.

Calculate the Velocity Change for the Earth

Using the conservation of momentum, compute the change in Earth’s velocity from the fusion with the comet. Use Earth’s mass (5.97 \( \times 10^{24} \) kg) to calculate \( \Delta v_{Earth} \).

Determine Notability of Changes

Assess each calculated change against typical comet and planetary speeds to determine if they are significant or negligible.

Key concepts

Collision Mechanics

In a collision, objects interact by exchanging forces over a short period of time. This exchange affects their velocities and possibly their directions.
Controlling factors in collisions include mass, initial velocities, and the nature of the collision (elastic or inelastic).
NASA’s Deep Impact involved inelastic collision mechanics, where the 372 kg probe collided with the comet. Here, momentum conservation is crucial: the momentum before the collision equals the momentum after.
For example, in inelastic collisions, objects may stick together after collision, losing some kinetic energy in the process, usually as heat or deformation.
By setting initial momentum \[ m_{comet} \times v_{comet} + m_{probe} \times v_{probe} = m_{comet} \times v_{comet,final} \] we calculate the post-impact velocity.
Understanding these mechanics helps assess impact effects.

Comet Impact

Comet impacts involve interactions of celestial bodies, offering insights into both collision mechanics and cosmic events.
This involves a comet, such as Tempel 1, encountering another celestial object, like NASA's probe. The kinetic energy and momentum from these impacts can reveal physical characteristics of the comet.

• Comet Tempel 1 was moving at 40,000 km/h.
• The probe, moving at 37,000 km/h, transferred its momentum to the comet upon impact.

The effects of such impacts can change the velocity of the body hit. However, given the massive size difference between the comet and the probe, the change in velocity may be small and difficult to notice without precise measurement tools. Furthermore, by calculating the momentum exchange, scientists can determine if such a change would have noticeable long-term effects on the comet's trajectory or rotation.

Velocity Change

When discussing velocity change, we're referring to the adjustment in speed and direction of a body post-collision. For the exercise, two scenarios are examined:

• The velocity change in the calculated final velocity of comet Tempel 1 post-probe impact.
• Potential velocity change of Earth if it collided with the comet.

For the comet:

• Using momentum conservation, the final velocity becomes slightly altered but not significantly noticeable in short-term observations.
• Change: \( \Delta v_{comet} = v_{comet,final} - v_{comet}\) suggests minute variations.

For the Earth:

• Massive comparison (comet's 0.10 \( \times 10^{14} \) kg to Earth's 5.97 \( \times 10^{24} \) kg) results in negligible velocity change.
• \( \Delta v_{Earth} \) is effectively unnoticeable, showcasing how momentum distributes differently across mass disparities.

Velocity changes become noticeable only when the masses and initial velocities are relatively comparable.

Earth-Motion Implications

Though comet collisions may seem catastrophic, their effects on Earth’s motion are usually minimal due to the massive size of our planet in comparison.
The theoretical collision with comet Tempel 1 serves as an illustration:

• The tremendous mass of Earth (5.97 \( \times 10^{24} \) kg) means any celestial body must be vastly massive to impact its velocity significantly.
• Momentum conservation calculations show a minuscule change in Earth’s velocity \( \Delta v_{Earth} \).

In conclusion, while the collision mechanics are applicable, the despite massive energy and force involved, changes in Earth's motion due to such impacts are hardly perceivable.
Most cosmic impacts, like small comets or asteroids, do not carry enough momentum to shift our planet's path or rotational characteristics. Understanding this helps contextualize comet impacts in terms of broader astrophysical phenomena and the maintenance of Earth’s relative stability.