Negative potential wells and escape speed

Science · Forces & Motion · ages 16-18

Name ______________________ Date ____________

With gravitational potential set to zero at infinity, the potential outside a spherical planet at any finite distance from its centre is negative.
Circle one: True False
Outside an isolated spherical mass, gravitational potential is written as V = -GM/r. Where does V approach zero?
- At infinity
- At the surface
- At the centre
How is escape speed found?
- Divide gravitational force by the object's mass
- Set kinetic energy equal to the well depth
- Multiply gravitational field strength by radius
A craft is lifted from the surface to orbit. What is the sign of its potential energy change?
- Positive, since moving outward raises potential energy
- Negative, since increasing distance lowers potential energy
- Zero, since an orbit has no gravitational potential energy
A 2 kg rock sits where potential is minus 5 J/kg. U equals m V. Give U in joules.
Answer: ______________
An object falls inward under gravity alone. Which statement describes its energies before any collision?
- U and K both decrease by equal amounts during the fall, so total mechanical energy E decreases.
- U decreases and K increases equally, so total mechanical energy E stays constant.
- U stays unchanged while K increases during the fall, so total mechanical energy E increases.
Two spherical planets have the same mass. Planet X has the smaller radius. Ignoring air drag and other bodies, which has the greater surface escape speed?
- The larger planet, because its greater radius deepens the well
- Both planets, because equal masses give equal surface potentials
- Planet X, because its smaller radius makes the well deeper
Take U = 0 at infinity. A satellite has negative total mechanical energy E = K + U relative to Earth. With Earth's gravity as the only force, it can escape to infinity.
Circle one: True False

Answer key

For grown-ups. Fold this page away before handing over the rest.

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True · With zero at infinity, V = -GM/r outside the planet. G, M and r are positive, so V is negative at every finite exterior distance.
At infinity · For positive G and M, -GM/r approaches zero as r grows without limit. Under the stated convention, V therefore approaches zero at infinity.
Set kinetic energy equal to the well depth · At the minimum escape speed, K equals -U. The object approaches infinity with both U and K approaching zero.
Positive, since moving outward raises potential energy · Moving farther from Earth's centre makes U = -GMm/r less negative. The final U is greater than the initial U, so the change is positive.
-10 · Multiply 2 by minus 5 to get minus 10 J.
U decreases and K increases equally, so total mechanical energy E stays constant. · During an inward fall, U becomes more negative. Gravity transfers that decrease into an equal increase in K, leaving E = K + U constant.
Planet X, because its smaller radius makes the well deeper · For equal M, v = sqrt(2GM/R) is greater when R is smaller. Planet X therefore has the greater surface escape speed.
False · Gravity alone conserves the negative E. As distance approaches infinity, U approaches zero and K cannot be negative. Escape would require E of at least zero, so the statement is false.