AP Physics C 2017

Proportional Reasoning

These also test your knowledge of how to use equations, except that you don’t have to plug in numerical
values to solve them.

Planet  X   is  twice   as  massive as  Earth,  but its radius  is  only    half    of  Earth’s radius. What    is  the

acceleration    due to  gravity on  Planet  X   in  terms   of  g   ,   the acceleration    due to  gravity on  Earth?

(A)     ¼   g

(B)     ½   g

(C)     g

(D)     4g

(E)     8g

First we need to know what equation to use. We know that the force that a planet exerts on a small
mass m 1 near its surface is

Using Newton’s second law (F (^) net = ma ), we know that the acceleration of the small mass is simply
One method of solution would be to plug in the actual mass and radius of the new planet. But no fair, you
say, the mass of the Earth isn’t given on the constants sheet. How do I find the mass of the planet?
You don’t!
Use proportional reasoning skills instead, so:
“The mass of the planet is twice that of the Earth. Since mass is in the numerator of the
equation for acceleration, doubling the mass of the planet must double the acceleration .
“Okay, but the radius of this planet is also different. Radius is in the denominator, so a
smaller radius means a bigger acceleration. The radius of the new planet is half of the radius of
the Earth. Therefore, the acceleration must be doubled. Almost there ... because the radius is
SQUARED in the denominator, the acceleration must be doubled AGAIN .
“So what is my final answer? The mass causes acceleration to double. The radius causes the
acceleration to double, and then to double again. So the total acceleration is multiplied by a
factor of 8. The acceleration on this planet is 8g.”
In the much more concise language of algebra, your reasoning might look like this: