An object is on a collision course with the Earth and is predicted to hit in the centre of the target, coming in vertically. The object is roughly spherical, with an approximate diameter of\(100\;{\rm{m}}\)(the meteor that damaged Chelyabinsk, Russia, in February 2013 had a diameter of about\(20\;{\rm{m}}\)the object that killed off the dinosaurs 65 million years ago is thought to have had a diameter of about\(10\;{\rm{km}}\)). If we start immediately, we can rendezvous with the object when it is\(\)\(2.5 \times {10^{11}}\;{\rm{m}}\)from the Sun, traveling toward the Sun at a speed of\(30\;\;{\rm{km/s}}\)There is a proposal to implant a rocket engine on the surface of the object to deflect the object enough to miss the Earth. Perform an approximate feasibility analysis of this critical mission. Give details of your design, including the estimates, assumptions, and idealizations you make. Note that the first stage of the most powerful rocket ever used, the Saturn V, had a thrust of\(3.4 \times {10^7}\;{\rm{N}}\)and burned for\(170\;{\rm{s}}\)Can the Earth be saved?

Given that the radius \(r\) of the object is \(50\;{\rm{m}}\), the net force \({F_{{\rm{net }}}}\) achieved by the rocket is \(3.4 \times {10^7}\;{\rm{N}}\), the time interval \(\Delta t\) which rocket burns is \(170\;{\rm{s}}\), assume that the initial velocity \({v_i}\) of the object is \(3 \times {10^5}\;{\rm{m}}/{\rm{s}}\) and assume that the density \(\rho \) of the rock is \(8 \times {10^3}\;{\rm{kg}}/{{\rm{m}}^3}\).

The quantity of motion that occurs in something that is moving, or the force that propels something forward to keep it moving, is defined as momentum.

The rate at which an automobile descends a hill is an illustration of momentum.

Formulae of momentum is\(p = mv\).

calculate the mass of this object,

\(\)\(\begin{aligned}{c}m &= \rho V\\ \approx \left( {7.5 \times {{10}^3}} \right)\left( {\frac{4}{3} \times \pi \times {{50}^3}} \right)\\ &= 3.93 \times {10^9}\;{\rm{kg}}\end{aligned}\)

Utilize the momentum principle when the rocket and the object collide, and the impulse will \(\)be

\(\begin{aligned}{c}{\rm{ Impulse }} &= F\Delta t\\ &= \left( {3.4 \times {{10}^7}} \right)(170)\\ &= 5.78 \times {10^9}\;\;{\rm{N}} \cdot {\rm{s}}\end{aligned}\)

Thus, the object will have a speed of

\(\begin{aligned}{c}{v_f} &= \frac{{{p_f}}}{m}\\ &= \frac{{1.18 \times {{10}^{14}}}}{{3.93 \times {{10}^9}}}\\ \approx 3 \times {10^4}\;{\rm{m/s}}\end{aligned}\)

The object will have a speed of \(3 \times {10^4}\;{\rm{m}}/{\rm{s}}\)

Thus, the objects' initial and ultimate speeds become comparable. Because the most powerful rocket ever fired has no effect on the object's speed, it will arrive at the point where the object collides with the ground at the same time. According to the assumptions made in the answer, the earth would not be preserved. As a result, it is uncertain.