The Problem¶

John is cruising down the highway at 30 $\frac{m}{s}$ when he rounds a curve and the sees a police officer scanning the speeds of drivers. Without thinking, and unaware that he was already below the speed limit, John taps the brakes, causing his car to decelerate at -6 $\frac{m}{s^2}$. If John holds his brakes for 1.2 seconds, how fast is he going when he finally lets up on the brake pedal?

The Answer¶

The secret to solving word problems is to G.U.E.S.S.!

G stands for Given. Write down all the information given that will help you solve the problem. This can include a picture, unit conversions if needed, or anything else in the problem that can help you solve it.
U stands for Unknown. Write down what the problem asks you to find. I like to write it like "$x = ?$"
E stands for Equation. Look at the Givens and the Unknown, and try to think of a formula or equation that you have enough information to fill out everything except the unknown.

Before we continue, take a look and notice that you have translated to word problem to a regular problem. Yay!

S and S stand for Substitute and Solve, but depending on the problem and your math skills, they don't have to be in that order.

Once you are done GUESSing, you will have the answer to your word problem. Now let's see how it works!

Given¶

Reading the probelem gives us this information:

$v_{i} = 30 \frac{\text{m}}{\text{s}}$ because "John is cruising down the highway at 30 meters per second"
$a = -6 \frac{\text{m}}{\text{s}^2}$ because "causing his car to decelerate at..."
$t = 1.2$s because " holds his brakes for 1.2 seconds"

Unknown¶

The question at the end of the problem usually tells us what the unknown is.

$v_f = ?$ because "how fast is he going when he finally lets up on the brake pedal?"

Equation¶

This problem is in the topic of 1-D Motion, or Kinematic Equations. There are only 3 equations relating to this topic; one equation relates our givens and unknown.

$v_f = v_i + a t$ ← This one!
$s = v_i t + \frac 1 2 a t^2$
$2 a s = v_f^2 - v_i^2$

The first equation contains all of our unkowns and givens. Sweet! We'll use that one.

SS...¶

Believe it or not, at this point the problem is pretty much solved. A little bit of Algebra 1 is all that remains. There are a couple of different ways this can be done...

Method 1: Substitute, then Solve¶

If you are uncomfortable manipulating an equation without all the numbers, you should probably substitute first. Rewrite the equation and "plug in" the values that were given. Then solve, probably using a calculator quite a bit. (Note that the units are all SI units, so they should work out.)

$$v_f = v_i + a t$$

$$v_f = 30 + (-6)\times 1.2$$ $$v_f = 30 - 7.2$$

$$v_f = 22.8 \frac{\text{m}}{\text{s}}$$

Method 2: Solve, then Substitute¶

If you can handle it, consider solving the symbolic equation before plugging in the numbers or grabbing a calculator. Since there are only a few equations and a few variables, there can only be a relatively small number of unique problems. Solving symbolically and then plugging in the givens is a good way to start noticing that, and also reduces your calculator usage.

Luckily, in this case, our equation is already solved for in terms of our unknown variable! So the solve then substitute method will look exactly the same as the substitute then solve method:

$$v_f = v_i + a t \text{ ← Already solved for $v_f$!}$$$$v_f = 30 + (-6)\times 1.2$$$$v_f = 30 - 7.2$$

$$v_f = 22.8 \frac{\text{m}}{\text{s}}$$

So John's final speed was 22.8 m/s, still well below the speed limit.

Physics 1: Problem of the Day Archive

Problem Daily Day 3¶

October 4, 2020¶