Question

NASA has taken an increased interest in near-Farth asteroids. These objects, popularized in certain blockbuster movies, can pass very close to Earth on a cosmic scale, sometimes as close as 1 million miles. Most are small-less than \(500 \mathrm{~m}\) across-and while an impact with one of the smaller ones could be dangerous, experts believe that it may not be catastrophic to the human race. One possible defense system against near-Earth asteroids involves hitting an incoming asteroid with a rocket to divert its course. Assume a relatively small asteroid with a mass of \(2.10 \cdot 10^{10} \mathrm{~kg}\) is traveling toward the Earth at a modest speed of \(12.0 \mathrm{~km} / \mathrm{s}\) a) How fast would a large rocket with a mass of \(8.00 \cdot 10^{4} \mathrm{~kg}\) have to be moving when it hit the asteroid head on in order to stop the asteroid? Assume that the rocket and the asteroid stick together after colliding. b) An alternative approach would be to divert the asteroid from its path by a small amount and thus cause it to miss Earth. How fast would the rocket of part (a) have to be traveling at impact to divert the asteroid's path by \(1.00^{\circ} ?\) In this case, assume that the rocket hits the asteroid while traveling along a line perpendicular to the asteroid's path.

Short answer

Answer: To stop the asteroid completely, the rocket must be moving at 31,500 km/s in the opposite direction of the asteroid. To divert the asteroid's path by 1°, the rocket must be moving at approximately 209 m/s along the line perpendicular to the asteroid's path.

Step-by-step solution

a) Stopping the asteroid

To stop the asteroid, we need to consider the conservation of linear momentum. The initial total momentum of the rocket and the asteroid must be equal to their final total momentum after the collision. Let's call the initial speed of the rocket just before the collision v_r. Then we have: Initial momentum = Final momentum \(M_a \cdot V_a + M_r \cdot V_r = (M_a + M_r) \cdot V_f\) Since we want to find the value of v_r that makes the asteroid stop, the final velocity of both the rocket and the asteroid will be 0. \(0 = M_a \cdot V_a + M_r \cdot V_r\) Solving for \(V_r\), we get: \(V_r = -\frac{M_a}{M_r} \cdot V_a\) Now substitute the given values: \(V_r = -\frac{2.10 \cdot 10^{10} \mathrm{~kg}}{8.00 \cdot 10^{4} \mathrm{~kg}} \cdot 12.0 \mathrm{~km/s} \) \(V_r = -31500 \mathrm{~km/s}\) So, the rocket must be moving at 31,500 km/s in the opposite direction of the asteroid to stop it completely.

b) Diverting the asteroid by 1°

In this part, we will consider the rocket hitting the asteroid perpendicularly. If the deviation angle is 1°, then the tangent of that angle is the ratio between the perpendicular velocities of the system after the collision. We can use the conservation of linear momentum in both the perpendicular and parallel direction of the asteroid's initial path. Let's call the perpendicular velocity of the rocket before the collision \(V_{rp}\). Then we have: \(M_a\cdot V_a = (M_a+M_r)\cdot V_x\) \(M_r\cdot V_{rp} = (M_a+M_r)\cdot V_y\) Dividing the second equation by the first: \(\frac{V_{rp}}{V_a}=\frac{V_y}{V_x}\) Since the asteroid's path is diverted by 1°, we have: \(\tan{(1°)}=\frac{V_y}{V_x}\) Now, replace \(\frac{V_y}{V_x}\) with \(\frac{V_{rp}}{V_a}\): \(V_{rp} = V_a \cdot \tan{(1°)}\) Now substitute the given values: \(V_{rp} = 12.0 \mathrm{~km/s} \cdot \tan{(1^{\circ})}\) \(V_{rp} \approx 209 \mathrm{~m/s}\) So, the rocket must be moving at approximately 209 m/s along the line perpendicular to the asteroid's path to divert its trajectory by 1°.

Key concepts

Inelastic Collisions

An inelastic collision, a fundamental concept in physics, refers to an event where colliding objects coalesce post-impact, ultimately moving together as a single unit. This type of collision is distinguished by the conservation of linear momentum; however, kinetic energy is not conserved.

During such an encounter, as explained in the exercise where a rocket collides with an asteroid, the two objects stick together. To analyze this scenario, physicists employ the conservation of momentum, which stipulates that the total momentum before the collision is equivalent to the total momentum after it. For instance, if a massive asteroid and a smaller rocket were to collide inelastically—assuming that the rocket halts the asteroid—the rocket's necessary velocity can be deduced using the formula:
\[ V_r = -\frac{M_a}{M_r} \cdot V_a \]

In this mathematical expression, \( M_a \) and \( M_r \) are the masses of the asteroid and rocket, respectively, while \( V_a \) and \( V_r \) represent their velocities. The negative sign indicates that the rocket must propel in the opposite direction to negate the momentum of the oncoming asteroid.

Momentum

The term 'momentum' describes the quantity of motion an object possesses and is directly relevant to inelastic collisions, among other types of physical interactions. Momentum, symbolized by the letter 'p,' is a vector quantity, meaning it possesses both magnitude and direction. It is calculated by multiplying an object's mass \( m \) by its velocity \( v \):
\[ p = m \cdot v \]

Conservation of linear momentum is a powerful principle in physics that asserts that within a closed system not subject to external forces, the total linear momentum remains constant. This rule is paramount when considering exercises like the asteroid deflection scenario, where the momentum of the system before and after impact must be equal. We expressed this in the original exercise solution by equating the momentum of the asteroid-rocket system before and after the event.

An appreciation for momentum is crucial, not only for solving textbook problems but also for understanding the behavior of objects in real-world phenomena, such as vehicular collisions or sports mechanics.

Asteroid Deflection

Asteroid deflection is a strategic concept in planetary defense, involving the diversion of a potentially hazardous asteroid's trajectory to prevent an impact with Earth. It is not only a topic of high interest for scientists and governments but also a real-world application of physics principles like momentum and energy conservation.

When contemplating asteroid deflection, experts consider various methods. A kinematic impact, akin to the exercise example, uses a spacecraft or rocket to transfer momentum to the asteroid, intending to minutely alter its course over time so that it eventually misses Earth.

In a scenario where the asteroid's trajectory needs to be changed by a small angle such as \(1^\circ\), as seen in part b of the exercise, the rocket's required velocity can be computed taking into account trigonometric relationships and conservation principles:
\[ V_{rp} = V_a \cdot \tan{(1^\circ)} \]

This equation represents the complementing perpendicular velocity needed by the rocket to achieve the desired deflection angle. Moreover, effective asteroid deflection strategies are influenced by factors such as the asteroid's size, composition, and the time available before potential impact.