Question

1. explain why it is so difficult to achieve escape velocity in a rocket.

Explanation:

Achieving escape velocity is difficult for rockets due to several factors. First, the high energy requirement: escape velocity \(v_{e}=\sqrt{\frac{2GM}{R}}\) (where \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(R\) is the radius of the planet). For Earth, \(v_{e}\approx 11.2\space km/s\). This means a rocket needs a large amount of kinetic energy \(K = \frac{1}{2}mv_{e}^{2}\) (where \(m\) is the mass of the rocket). Second, the rocket has to work against Earth's gravitational force throughout its ascent. As it moves away from the Earth, the gravitational force \(F=\frac{GMm}{r^{2}}\) (where \(r\) is the distance from the center of the Earth) still acts on it, requiring continuous thrust. Third, air resistance: in the lower atmosphere, air resistance \(F_{d}=\frac{1}{2}
ho v^{2}CA\) (where \(
ho\) is the air density, \(v\) is the velocity of the rocket, \(C\) is the drag coefficient, and \(A\) is the cross - sectional area) acts against the motion of the rocket, dissipating energy as heat. Fourth, the mass of the rocket itself: a large rocket has a large mass \(m\), and from \(F = ma\) (Newton's second law), a larger force \(F\) is required to accelerate it to the high escape velocity. Also, the rocket needs to carry a large amount of fuel, which adds to its mass, creating a complex mass - fuel - thrust relationship.

Answer:

Achieving escape velocity is difficult for rockets due to high energy requirements (based on \(v_{e}=\sqrt{\frac{2GM}{R}}\) and \(K=\frac{1}{2}mv_{e}^{2}\)), work against gravity (\(F = \frac{GMm}{r^{2}}\)), air resistance (\(F_{d}=\frac{1}{2}
ho v^{2}CA\)), and mass - thrust considerations (from \(F=ma\)).