The Math Behind Motion: Braking at 30 Mph vs 60 Mph

     This post will be quite simple.  We’ll work with only 1 equation here to show you the effects of increasing our speed from 30 mph to 60 mph in relation to braking.  The energy in question is Kinetic Energy (J), or energy that’s in motion.  Since our car is moving, it has a certain amount of kinetic energy.  But just how much?  And how does it compare/matter when looking at the effects of braking/hitting something.  We’ll be using two variables here; Mass (kg) and Velocity (m/s):

Kinetic Energy (J) = Kinetic Energy

A basic physics equation, KE = mv2/2 where we square Velocity, then multiply that by Mass and then divide by 2 to get our Kinetic Energy.

     The focus of this post is the (Velocity)2.  Our Velocity is squared, no matter how slow/fast we’re going, so our Kinetic Energy will also end up being exponential, not linear.  When our speed is doubled, our Kinetic Energy is quadroupled here.  If you can imagine the graph, it would be a parabola starting at the origin, and curving upwards.

     So lets start plugging in numbers.  For reference, we’ll use a 3,000 pound car at speeds of 30 mph and 60 mph.  We’ll first need to convert our Imperial weight to Metric, which equals to approximately 1361 kgs.  Then convert our Imperial speed of 30 mph to Metric, which equals to approximately 13.4 m/s.  Now we plug everything in:

Kinetic Energy (J) = Kinetic Energy Example

Kinetic Energy (J) ≈ 122,000 Joules

     This 122,000 joules is the amount of kinetic energy the 3,000 pound car has when traveling at 30 miles per hour.  That is also the amount of energy needed to be converted to heat energy by the brakes to bring a car to a full stop.  So if we’re thinking from a braking perspective, obviously, the slower we are, the less energy is needed to bring the car to a stop.  Now let’s plug in 60 mph for the equation:

Kinetic Energy (J) = Kinetic Energy Example2

Kinetic Energy (J) ≈ 488,000 Joules

     We can see here that just by doubling our speed, our brakes have to work four times harder.  Also, you’ve probably noticed, but lessening the mass here also can lessen the work load on the braking system.  For reference, 122,000 Joules is the equivalent of exploding 30 grams of TNT, while 488,000 Joules would be exploding 120 grams of TNT.

     So the important lesson here is that kinetic energy is not linear, but exponential.  But the other important part of the equation is the Mass.  As Mass increases, so does our Kinetic Energy.  As our Mass decreases, so does our Kinetic Energy.  And when it comes to braking, we want to stop as fast as possible.  The quickest way to this solution (other than slowing down) is to lessen our Mass.  That’s important when building a sports car.  The less weight, the less work the car needs to do.  And if the car doesn’t need to work as much, we don’t need all the equipment of powering a 4,000 pound car.  So we shed more weight there, and the cycle continues.

     Now lets take a look at how the Stopping Distance (ft) is affected.  We need two variables here, the vehicle’s Velocity (mph) and Deceleration (g).  Then just plug it into this equation:

Stopping Distance (ft) = Stopping Distance

All we’re doing here is squaring our Velocity, and then dividing by the product of the Deceleration (in g(s)) and 29.9 (a conversion factor).

     Once again, we see that our vehicle speed has an exponential affect on our result.  If we plug in 30 mph and 60 mph for our Velocity, lets see what we get for our stopping distances (lets use 1 g as our deceleration):

Stopping Distance (ft) = Stopping Distance Example

Stopping Distance (ft) ≈ 30 feet

Stopping Distance (ft) = Stopping Distance Example2

Stopping Distance (ft) ≈ 120 feet

     So we can see here that once again, by doubling our speed, we quadrouple our stopping distance.  Now imagine an emergency stop at 30 mphs and 60 mphs.  That’s a difference of 90 feet.  While barely stopping before another car at 30 mph is reasonable, there’d be no way you’d be able to stop at 6o mph and avoid the same person.  Their car would take around 30 feet to stop, while you would require an extra 90 feet.  Even if you were 5 car lengths behind him/her (using an average of 17 feet for an average car), you still wouldn’t have enough distance.  Factor in the time between when her car starts braking and the time you start braking, you’ll have traveled another 10 feet or so.  I don’t want to come off as preaching here, but I’m just here to inform you.  Read what you want to read, and take away what you will, but this information here is for everyone’s access.  Enjoy driving, but keep it safe.


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April 2009
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© 2009 Rusi Li

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