Q:

Two metal spheres are the same size but have different masses. They are dropped simultaneously from a cliff. How do their accelerations compare at midpoint in their fall? Does one mass reach the midpoint before the other?

- Anonymous

Beck JH

- Anonymous

Beck JH

A:

In your school, you’ve probably heard about gravity. Gravity is what pulls everything toward the center of the Earth. The cool thing about gravity is that (in the absence of other forces) it causes everything to have the same acceleration, no matter how heavy it is.

The reason for this is can be seen by looking at Newton’s 2nd law: "F = ma". In the case of gravity near the earths surface, the force on an object that has a mass "m" is "mg", that is, the mass of the object times the constant "g". So if we replace "F" by "mg" Newton’s law becomes "mg = ma". We can divide this by "m" to get "g = a", that is, the acceleration "a" is the same as the constant "g".

If you ignore air resistance (see below), then the spheres will fall exactly side by side as they fall, reaching the midpoint at the same time, and then hitting the ground at the same time.

What if we don’t ignore the effect of the air? I’m sure that you’ll notice if you wave your hand in the air, or if you’ve ever seen a dog sticking its head out a car window while driving, you can see the air moving the dog’s fur or feel the air on your hand. Air also pushes on things - and the amount of this push depends on the size and shape of the object.

As an object falls, the air will push up on it to slow its fall a bit (this is called air resistance). The amount of air resistance on a falling object depends on how fast it is falling and on the shape of the object. Because the metal spheres in your problem are the same size and shape, the air resistance force on them will be the same (if they are falling at the same speed). Lets call the force of air resistance "r", and say that at some moment it’s the same for both spheres:

Using Newton’s law again: "F = ma". Now we have the force of gravity "mg" trying to speed the object up and the force "r" trying to slow it down. The total force is therefore "mg-r" so: "mg-r = ma" Now if we solve for "a" we get "a = g - r/m". If the mass is really big, the second term on the right hand side of the equal sign "r/m" will be small compared to "g" which means that we can ignore it and write "a = g" just like when there is no air to worry about. If the mass is small, we can’t ignore the "r/m" part, and the acceleration "a" will be smaller than "g" since "a = g - r/m". So, the answer to your question is that if you take air resistance into account, then the heavier sphere will reach the half-way point first.

-Sara & Mats

The reason for this is can be seen by looking at Newton’s 2nd law: "F = ma". In the case of gravity near the earths surface, the force on an object that has a mass "m" is "mg", that is, the mass of the object times the constant "g". So if we replace "F" by "mg" Newton’s law becomes "mg = ma". We can divide this by "m" to get "g = a", that is, the acceleration "a" is the same as the constant "g".

If you ignore air resistance (see below), then the spheres will fall exactly side by side as they fall, reaching the midpoint at the same time, and then hitting the ground at the same time.

What if we don’t ignore the effect of the air? I’m sure that you’ll notice if you wave your hand in the air, or if you’ve ever seen a dog sticking its head out a car window while driving, you can see the air moving the dog’s fur or feel the air on your hand. Air also pushes on things - and the amount of this push depends on the size and shape of the object.

As an object falls, the air will push up on it to slow its fall a bit (this is called air resistance). The amount of air resistance on a falling object depends on how fast it is falling and on the shape of the object. Because the metal spheres in your problem are the same size and shape, the air resistance force on them will be the same (if they are falling at the same speed). Lets call the force of air resistance "r", and say that at some moment it’s the same for both spheres:

Using Newton’s law again: "F = ma". Now we have the force of gravity "mg" trying to speed the object up and the force "r" trying to slow it down. The total force is therefore "mg-r" so: "mg-r = ma" Now if we solve for "a" we get "a = g - r/m". If the mass is really big, the second term on the right hand side of the equal sign "r/m" will be small compared to "g" which means that we can ignore it and write "a = g" just like when there is no air to worry about. If the mass is small, we can’t ignore the "r/m" part, and the acceleration "a" will be smaller than "g" since "a = g - r/m". So, the answer to your question is that if you take air resistance into account, then the heavier sphere will reach the half-way point first.

-Sara & Mats

*(published on 10/22/2007)*