Question : (a) Estimate your average acceleration as you drive up the e...

This problem involves more than our usual amount of estimating! We are trying to come up with a value of a_{x} , but that value is hard to guess directly. The other three variables involved in kinematics are position, velocity, and time. Velocity is probably the easiest one to approximate. Let us assume a fin al velocity of 100 \mathrm{~km} / \mathrm{h}, so that you can merge with traffic. We multiply this value by 1000 to convert kilome-ters to meters and then divide by 3600 to convert hours to seconds. These two calculations together are roughly equivalent to dividing by 3 . In fact, let us just say that the fin al velocity is v_{x f} \approx 30 \mathrm{~m} / \mathrm{s} . (Remember, you can get away with this type of approximation and with dropping digits when performing men tal calculations. If you were starting with British units, you could approximate 1 \mathrm{mi} / \mathrm{h} as roughly 0.5 \mathrm{~m} / \mathrm{s} and contin ue from there. Now we assume that you started up the ramp at about onethird your final velocity, so that v_{x i} \approx 10 \mathrm{~m} / \mathrm{s} . Finally, we assume that it takes about 10 \mathrm{~s} to get from v_{x i} to v_{x f} , basing th is guess on our previous experience in automobiles. We can then find the acceleration, using Equation 2.8 :

Granted, we made many approximations along the way, but this type of mental effort can be surprisingly useful and often yields results that are not too different from those derived from careful measurements.

(b) How far did you go during the first half of the time interval during wh ich you accelerated?

We can calculate the distance traveled during the first 5 \mathrm{~s} from Equation 2.11

x_{f}-x_{i} =v_{x j} t+\frac{1}{2} a_{x} t^{2} \approx(10 \mathrm{~m} / \mathrm{s})(5 \mathrm{~s})+\frac{1}{2}\left(2 \mathrm{~m} / \mathrm{s}^{2}\right)(5 \mathrm{~s})^{2}
=50 \mathrm{~m}+25 \mathrm{~m}=75 \mathrm{~m}

This result indicates that if you had not accelerated, your initial velocity of 10 \mathrm{~m} / \mathrm{s} would have resulted in a 50 \mathrm{~m} movement up the ramp during the first 5 \mathrm{~s} . The additional 25 \mathrm{~m} is the result of your increasing velocity during that interval.Do not be afraid to attempt making educated guesses and doing some fairly drastic number rounding to simplify mental calculations. Physicists engage in this type of thought an alysis all the time