Estimate the magnitude of the momentum of a runner in a marathon.
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In this problem, you are asked to estimate momentum in a single situation. Although no information is given, a quick look at the definition of momentum shows that it depends only on mass and velocity—two quantities that you can estimate. So you are not asked to calculate new information in this problem but rather to restate what you can say about motion in a slightly different way.
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There is no need for a picture in most definition problems, and this is one of them. You can estimate velocity information for a situation and asked for the closely-related momentum information for the same motion. A picture will not provide any additional insight or organization beyond what is already present in the problem.
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In equation form, momentum is defined as
p = mv
This is the only relation you need for this problem. -
p = mv
p = mv
p ≈ (55 kg)(4 m/s)
p ≈ 200 kg m/s
There is no further calculation required in this problem.
You are only asked for the magnitude of momentum in this problem.
You are asked to estimate values for mass and velocity, and so your answer gives an approximate value for momentum.
Runners tend to be lean, so I estimated a lower than average mass. This corresponds to a weight of about 120 lb.
The symbol for momentum is p. The bold type reminds you that momentum is a vector, as is velocity.
A quick on-line search shows that a marathon is 42,195 m long and good times are in the neighborhood of 3 hours or a little more.
(42,195 m)/(3 hr)(3600 s/hr) = 4 m/s
4 m/s is about 9 mph. Another on-line search shows that people at the gym tend to set treadmills at about 7 mph, so this is a reasonable number.The units of momentum are those of mass x velocity. There is no abbreviated unit such as a Newton or Joule.
Because I only estimated velocity to one significant figure, I only kept one significant figure in the answer.
How do you know the units for momentum?
The units of momentum are those of mass x velocity. There is no abbreviated unit such as a Newton or Joule.
Because I only estimated velocity to one significant figure, I only kept one significant figure in the answer.
Why do you call this a definition problem rather than a Conservation of Momentum problem?
Conservation of momentum allows you to compare two different points in a motion and learn something about that motion through the comparison. It is useful for understanding a situation in which momentum is transferred between objects within a system (i.e. when objects collide or when one object separates into pieces.) In this case, you are not asked for how momentum changes, but rather for its value in a single situation.
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p = mv
p = mv
p ≈ (55 kg)(4 m/s)
p ≈ 200 kg m/s
This problem is merely a definition problem. You will use the definition of momentum in Conservation of Momentum problems, much as you use Fg = mg as you solve force problems. Whenever you have information about the speed of an object, you can restate that information as its momentum (or vice versa.)
Your estimated value of momentum may be higher or lower depending on the values you used for mass and velocity. You should make sure that those values are in a reasonable range.