A tennis player places a 40 kg ball machine on a frictionless surface.

Question

The machine fires a 0.057 kg tennis ball horizontally with a velocity of 32 m/s toward the north. What is the final velocity of the machine?

Answer 1

This question can seem intimidating, with its numbers and descriptions -- but we can break it down to something very simple. The first step in any physics problem is to recognize the concepts in play. For instance, there is a word that appears nowhere in this problem -- but is utterly fundamental to finding its solution. That word is momentum.

Momentum seems like an easy thing to calculate -- just "multiply an object's mass by its velocity." That ease belies its importance, which comes from the law of conservation of momentum. This is the less famous sister of the law of conservation of energy, which says energy cannot be created or destroyed (but can be switched into and out of various forms, including matter).

Likewise, the momentum of object(s) must always sum to the same value at any time. But unlike energy, momentum is not just a value. It's calculated from velocity, and both are thus vectors -- they have a direction in addition to a value. And that means that unlike energy, parts of a system can wind up with more momentum than they had before... as long as they cancel out by going in opposite directions.

That's why it's important for this question to mention that the ball is fired by the machine to the north. Beforehand, the machine is not moving, and has no momentum. But after it's fired, the ball clearly does have momentum as it heads north. Since momentum is conserved, where did that momentum come from? It has to be balanced... by an equal and opposing amount of momentum pointing south. Something else must have gained that southward momentum... like the machine that fired the ball.

Now we have an organizing concept! We know that the ball has a northward momentum, which can be found from its mass and velocity. And we can deduce from the conservation of momentum that the machine will have a southward momentum of exactly the same size. Or, to put all this in mathematical symbols,

ν_ball + ν_machine = 0

The greek letter 'ν' or 'nu' is the standard symbol for momentum. The total momentum started at zero, and so it must still be zero.

( m_ball × v_ball ) + ( m_machine × v_machine) = 0

Each object's momentum is composed of mass multiplied by velocity...

( m_ball × v_ball ) = -( m_machine × v_machine)

...and by the law of conservation of momentum, the ball's and machine's momenta must be equal and in opposite directions.

( m_ball × v_ball ) = m_machine × (-v_machine)

Here, the ball's northward velocity is represented by a positive value, and the machine's southward velocity by a negative value. (If you have to deal with more than one dimension, the math gets more complicated, but we don't have to bother for this problem.)

The question asks for the machine's velocity, so let's use algebra to get a formula for its value:

( m_ball × v_ball ) / m_machine = (-v_machine)

-( m_ball × v_ball ) / m_machine = v_machine

Hooray! Starting from nothing but a definition and a conservation law, we have built an equation to give us exactly the value we seek. It's finally time to deal with those numbers: pull out your calculator (app or gadget) and drop the values into the equation...

-( 0.057 kg × 32 m/s ) / 40 kg = v_machine

... and as my favorite teacher said, "plug and chug" to solve the equation:

-( 0.046 m/s ) = v_machine

Why not 0.0456 m/s? That's what my calculator says.

True, and if the question gave more exact values for every number (40.0 kg and 32.0 m/s), that would be the answer. But any calculation's output is only as good as its input. Since we used values with only two significant digits (digits that convey the value, rather than just acting as spacers), we can't report an answer with more than two significant digits.

Okay, so the answer's -0.046 m/s?

Sort of. Remember, we were using positive and negative to represent the north and south directions. So the full answer should write that representation out explicitly: the machine's velocity is 0.046 m/s to the south.