# Worksheet: Escape Speed

In this worksheet, we will practice calculating the vertical speed that an object must have in order to escape the gravitational pull of a planet.

**Q1: **

Earth has a mass of kg and a radius of 6,370 km. Calculate the escape speed of Earth, giving your answer to 3 significant figures.

**Q2: **

Triton is the largest moon of Neptune with a mass of kg and a radius of 1,350 km. What is the escape speed of Triton? Give your answer to 3 significant figures.

**Q3: **

Earth has a mass of kg and a radius of 6,370 km. Calculate the ratio of the escape speed on the surface of Earth to the escape speed 20 km above the surface of Earth. Give your answer to 6 significant figures.

**Q4: **

Earth has a mass of kg and a radius of 6,370 km. Mars has a mass of kg and a radius of 3,390 kg. Calculate the ratio of the escape speed on the surface of Mars to the escape speed on the surface of Earth. Give your answer to 3 significant figures.

**Q5: **

Europa is a moon of Jupiter that has a mass of kg and a radius of 1,560 km. At what speed would an object have to be launched vertically off the surface of Europa in order to escape its gravity? Give your answer to 3 significant figures.

**Q6: **

An object with a mass of 10 kg is launched vertically from the surface of Earth. Earth has an escape speed of . The object is given an initial speed of . The graph shows how the velocity of the object changes over time after it is launched.

Which of the labeled points on the graph marks the time the object is launched?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

What is the launch speed of the object? Give your answer to 3 significant figures.

Which of the labeled points on the graph marks the time the object reaches its greatest height above the surface of Earth?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

Which of the labeled points on the graph marks the moment just before the object hits the ground again?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

**Q7: **

Each of the following four graphs shows how the gravitational potential energy and the kinetic energy change over time for an object with a mass of 10 kg that is launched vertically from the surface of Earth at different speeds.

Which graph corresponds to the object with the greatest initial kinetic energy?

- Aiv
- Bii
- Ci
- Diii

Which graph corresponds to the object with the least initial kinetic energy?

- Aiv
- Bi
- Cii
- Diii

Which graph, if any, corresponds to objects that had an initial kinetic energy that was greater than the magnitude of their initial gravitational potential energy?

- Ai
- Biv
- Cii
- Diii

Which graph, if any, corresponds to objects that had launch speeds greater than the escape speed of Earth?

- Aii
- Biv
- Ci
- Diii

**Q8: **

The graph shows the masses and radii of five planets. Each planet is a perfect sphere with a constant density, but the density of each planet is different.

Which planet has the greatest escape velocity?

- A
- B
- C
- D
- E

Which planet has the lowest escape velocity?

- A
- B
- C
- D
- E

**Q9: **

The figure shows five angles at which an object could be launched from the surface of a planet.

For which angle would the launch speed required for the object to escape the planet’s gravity be lowest?

- A
- B
- C
- D
- E

**Q10: **

An object with a mass of 10 kg is launched vertically from the surface of Earth. Earth has an escape speed of . The object is given an initial speed of . The graph shows how the radial distance, , between the object and the center of Earth changes over time after it is launched.

Which of the labeled points on the graph marks the time the object is launched?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

Which of the labeled points on the graph marks the time the object reaches its greatest height above the surface of Earth?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

At which of the labeled points on the graph is the speed of the object instantaneously zero?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

Is the speed of the object slower at point or point ?

- APoint
- BPoint

**Q11: **

Each of the four graphs shows how the gravitational potential energy and the kinetic energy change over time for an object with a mass of 10 kg that is launched vertically from the surface of Earth at different speeds.

Which graph corresponds to the object with the greatest launch speed?

- Aii
- Bi
- Ciii
- Div

Which graph corresponds to the object with the slowest launch speed?

- Aii
- Biii
- Civ
- Di

Which graph corresponds to the object that takes the greatest amount of time to hit the ground again?

- Aii
- Biv
- Ciii
- Di

Which graph corresponds to the object that takes the least amount of time to hit the ground again?

- Aiv
- Bi
- Ciii
- Dii

**Q12: **

A small moon has a mass of kg and a radius of 473 km. At what speed would an object with a mass of 12 kg have to be launched vertically off the surface of the moon in order to escape its gravity? Give your answer to 3 significant figures.

**Q13: **

An object with a mass of 10 kg is launched vertically from the surface of Earth. Earth has an escape speed of . The object is given an initial speed of . The graph shows how the radial distance, , between the object and the center of Earth changes over time after it is launched.

Which of the labeled points on the graph marks the time the object is launched?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

Which of the labeled points on the graph marks the time the object reaches its greatest height above the surface of Earth?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

At which of the labeled points on the graph is the speed of the object instantaneously zero?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

Is the speed of the object slower at point or point ?

- APoint
- BPoint

**Q14: **

Figures A and B show the velocity over time of an object with a mass of 10 kg that is launched vertically at different speeds from the surface of Earth. Earth has an escape speed of . In figure A, the object is launched with a speed of . In figure B, the object is launched with a speed of .

Figure A

Figure B

Which of the following correctly explains why the velocity line is straight in figure A but curved in figure B?

- AThe atmospheric drag on an object is proportional to its speed, so as the speed becomes slower, the deceleration due to drag decreases.
- BThe greater the launch speed of the object, the higher into the atmosphere it reaches. The atmosphere is less dense higher up, so as the object gets higher, the deceleration due to drag on it decreases.
- CFurther away from the center of mass of Earth, the acceleration due to the gravitational pull of Earth is lower. The greater the initial launch speed of the object, the further away from Earth it is when it reaches its greatest height above Earth’s surface. When the object is launched closer to the escape velocity of Earth, it reaches heights in the middle of its trajectory where its acceleration is lower.

**Q15: **

An object is launched vertically from the surface of Earth with a speed . Earth has an escape speed of . The graph shows the gravitational potential energy and kinetic energy of the object over time after it is launched. Which of the following inequalities is true for ?

- A
- B
- C

**Q16: **

An object with a mass of 10 kg is launched vertically from the surface of Earth. Earth has an escape speed of . The object is given an initial speed of . The graph shows how the acceleration due to gravity, , that the object experiences changes over time after it is launched.

Which of the labeled points on the graph marks the time the object is launched?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

At which of the labeled points on the graph is the speed of the object instantaneously zero?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

Which of the labeled points on the graph marks the time the object hits the ground again?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

**Q17: **

The graph shows the masses and radii of five planets. Each planet is a perfect sphere, and they all have the same constant density.

Which planet has the greatest escape speed?

- APlanet B
- BPlanet C
- CPlanet E
- DPlanet A
- EPlanet D

Which planet has the lowest escape speed?

- APlanet C
- BPlanet A
- CPlanet E
- DPlanet B
- EPlanet D

**Q18: **

An object with a mass of 10 kg is launched vertically from the surface of Earth. Earth has an escape speed of . The object is given an initial speed of . The graph shows how the velocity of the object changes over time after it is launched.

Which of the labeled points on the graph marks the time the object is launched?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

What is the launch speed of the object? Give your answer to 3 significant figures.

- APoint
- BPoint
- CPoint
- DPoint
- EPoint

What is the speed of the object when it is at its greatest height above the surface of Earth?

Is the speed of the object slower at point or point ?

- APoint
- BPoint

Which of the labeled points on the graph marks the moment just before the object hits the ground again?

- APoint
- BPoint
- CPoint
- DPoint
- EPoint