Gravitational Energy Changes | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

Gravitational Energy Changes

First calculate final potential energy minus initial potential energy.

Then use energy conservation to connect the potential-energy change to kinetic energy or supplied energy.

Calculate energy changes

Moving out makes potential energy increase towards zero. Therefore final minus initial is positive, and energy must be supplied.

Potential-energy change means final value minus initial value. Its unit is J.
$\Delta E_P = E_{P,\text{ final}} - E_{P,\text{ initial}}$
If gravity is the only force doing work, kinetic energy $E_K$ plus potential energy $E_P$ stays constant. Both are measured in J.
$E_K+E_P=\text{constant}$
$\Delta E_K=-\Delta E_P$
When both radii have numerical values, write both negative energy values before subtracting.
$\Delta E_P=GMm\left(\dfrac{1}{r_i}-\dfrac{1}{r_f}\right)$
Moving out (r bigger): $E_P$ rises toward zero, so $\Delta E_P$ is positive and you must supply energy.
Moving in (r smaller): $E_P$ drops, so potential energy changes into kinetic energy.

Worked example

Find the energy change as a rock falls

A rock of mass 200 kg starts far away (at rest) and falls onto a planet of mass 4.9 × 10²³ kg and radius 3.4 × 10⁶ m. Find its change in gravitational potential energy and its speed at the surface.

Potential energy is zero at infinity and negative at the surface. Calculate the change.
$\Delta E_P = -\dfrac{GMm}{R}$
$\Delta E_P=-1.9 \times 10^{9} \text{ J}$
There is no air resistance, so the kinetic-energy gain has the same size as the potential-energy loss.
$\Delta E_K = -\Delta E_P$
$\Delta E_K=+1.9 \times 10^{9} \text{ J}$
Use the kinetic energy to find the speed.
$\tfrac{1}{2}mv^2 = 1.9 \times 10^{9}$
$v = 4.4 \times 10^{3} \text{ m s}^{-1}$

Potential-energy change:

Answer

-1.9\times10^9\ \text{J}

Kinetic-energy change:

Answer

+1.9\times10^9\ \text{J}

Speed at the surface:

Answer

4.4\times10^3\ \text{m s}^{-1}

The mass cancels in the speed calculation. Objects have the same speed at the surface if they start at the same distance with the same initial speed and there is no resistance.

Your turnquick check

An object falls inward under gravity with no resistance. What are the signs of $\Delta E_P$ and $\Delta E_K$ ?

Show worked answer

$\Delta E_P$ is negative and $\Delta E_K$ is positive. Their sum is zero when gravity is the only force doing work.

Gravitational Energy Changes

First calculate final potential energy minus initial potential energy.

Then use energy conservation to connect the potential-energy change to kinetic energy or supplied energy.

Calculate energy changes

Moving out makes potential energy increase towards zero. Therefore final minus initial is positive, and energy must be supplied.

Potential-energy change means final value minus initial value. Its unit is J.
$\Delta E_P = E_{P,\text{ final}} - E_{P,\text{ initial}}$
If gravity is the only force doing work, kinetic energy $E_K$ plus potential energy $E_P$ stays constant. Both are measured in J.
$E_K+E_P=\text{constant}$
$\Delta E_K=-\Delta E_P$
When both radii have numerical values, write both negative energy values before subtracting.
$\Delta E_P=GMm\left(\dfrac{1}{r_i}-\dfrac{1}{r_f}\right)$
Moving out (r bigger): $E_P$ rises toward zero, so $\Delta E_P$ is positive and you must supply energy.
Moving in (r smaller): $E_P$ drops, so potential energy changes into kinetic energy.

Worked example

Find the energy change as a rock falls

Potential energy is zero at infinity and negative at the surface. Calculate the change.
$\Delta E_P = -\dfrac{GMm}{R}$
$\Delta E_P=-1.9 \times 10^{9} \text{ J}$
There is no air resistance, so the kinetic-energy gain has the same size as the potential-energy loss.
$\Delta E_K = -\Delta E_P$
$\Delta E_K=+1.9 \times 10^{9} \text{ J}$
Use the kinetic energy to find the speed.
$\tfrac{1}{2}mv^2 = 1.9 \times 10^{9}$
$v = 4.4 \times 10^{3} \text{ m s}^{-1}$

Potential-energy change:

Answer

-1.9\times10^9\ \text{J}

Kinetic-energy change:

Answer

+1.9\times10^9\ \text{J}

Speed at the surface:

Answer

4.4\times10^3\ \text{m s}^{-1}

The mass cancels in the speed calculation. Objects have the same speed at the surface if they start at the same distance with the same initial speed and there is no resistance.

Your turnquick check

An object falls inward under gravity with no resistance. What are the signs of $\Delta E_P$ and $\Delta E_K$ ?

Show worked answer

$\Delta E_P$ is negative and $\Delta E_K$ is positive. Their sum is zero when gravity is the only force doing work.