Escape Speed - UNSOLVED PRACTICE SET

Q7. Define escape velocity. Derive the expression for escape velocity from a planet of mass M and radius R.

Q8. Why does escape velocity not depend on the mass of the body being projected? Does it depend on the angle of projection?

Q9. Calculate the escape velocity from the Moon. (M = 7.4 × 10²² kg, R = 1.74 × 10⁶ m)

Q10. In your school, a student asks: "If I jump hard enough, can I escape Earth?" Calculate the initial speed required for a 50 kg student to escape Earth and comment on the feasibility.

Q11. The escape velocity from Jupiter is about 60 km/s. Explain why Jupiter has such a high escape velocity compared to Earth.

Q12. What is the relationship between escape velocity and orbital velocity? Show that vₑ = √2 vₒ.

Q13. Derive the expression for escape velocity from a planet. Discuss how it depends on:

(i) Mass of the planet

(ii) Radius of the planet

(iii) Density of the planet

Show that for a planet of given density, vₑ ∝ R. Calculate the escape velocity from:

(a) Earth

(b) Mars (M = 6.4 × 10²³ kg, R = 3.4 × 10⁶ m)

(c) Jupiter (M = 1.9 × 10²⁷ kg, R = 7.1 × 10⁷ m)

Q14. A body is projected vertically upward from Earth's surface with velocity v.

(a) If v = 0.5vₑ, calculate the maximum height reached.

(b) If v = vₑ, show that the body reaches infinity with zero speed.

(c) If v = 2vₑ, calculate the speed at infinity.

(d) For v = 0.5vₑ, calculate the total energy of the body.

(e) Discuss the nature of the orbit for v < vₑ, v = vₑ, and v > vₑ.

Q15. Analyse the concept of escape velocity in the context of black holes:

(a) What happens to escape velocity as the mass of a body increases?

(b) What happens as the radius decreases?

(c) Define the Schwarzschild radius.

(d) Show that if a body of mass M is compressed within radius 2GM/c², the escape velocity exceeds c.

(e) Explain why nothing, not even light, can escape a black hole.

Q16. A rocket of mass 1000 kg is to be launched from Earth.

(a) Calculate the minimum kinetic energy required for the rocket to escape Earth's gravity.

(b) If the rocket is launched at 15 km/s, calculate its kinetic energy at infinity.

(c) Calculate the maximum height reached if launched at 8 km/s.

(d) What percentage of the launch energy is "wasted" if launched at exactly escape velocity?

(e) Discuss why rockets are usually launched eastward from locations near the equator.

Q17. The escape velocity from a planet is found to be 10 km/s. A satellite orbits the planet at a height where the orbital speed is 6 km/s.

(a) Calculate the escape velocity from the satellite's orbit.

(b) Calculate the minimum additional speed the satellite needs to escape.

(c) If the satellite's engines provide an impulse that increases its speed by 20%, will it escape?

(d) Calculate the total energy of the satellite in orbit.

(e) Discuss the energy requirements for different mission profiles.

Q18. In a school science fair, students build a model to demonstrate escape velocity using a ball and a curved track.

(a) Design an experiment to show that a minimum speed is needed for the ball to complete a vertical loop.

(b) Calculate this minimum speed for a loop of radius 20 cm.

(c) Relate this to the concept of escape velocity.

(d) What happens if the ball is released from a lower height?

(e) How would you modify the experiment to better simulate planetary escape?