(A) Estimate your average acceleration as you drive up the entrance ramp to an interstate highway.

Question

(B) How far did you go during the first half of the time interval during which you accelerated?

Accepted Answer

This problem involves more than our usual amount of estimating! We are trying to come up with a value of [latex] a_{x} [/latex] , but that value is hard to guess directly. The other variables involved in kinematics are position, velocity, and time. Velocity is probably the easiest one to approximate. Let us assume a final velocity of 100 km/h, so that you can merge with traffic. We multiply this value by (1000 m/1 km) to convert kilometers to meters and then multiply by (1 h/3 600 s) to convert hours to seconds. These two calculations together are roughly equivalent to dividing by 3. In fact, let us just say that the final velocity is [latex] v_{x f} \approx 30 m / s [/latex]. (Remember, this type of approximation and the dropping of digits when performing estimations is okay. If you were starting with U.S. customary units, you could approximate 1 mi/h as roughly 0.5 m/s and continue from there.)

Now we assume that you started up the ramp at about one third your final velocity, so that [latex] v_{x i} \approx 10 m / s [/latex]. Finally, we assume that it takes about 10 s to accelerate from [latex] v_{x i} ext { to } v_{x f} [/latex] , basing this guess on our previous experience in automobiles. We can then find the average acceleration, using Equation 2.6:

[latex] \bar{a}_{x} \equiv \frac{\Delta v_{x}}{\Delta t}=\frac{v_{x f}-v_{x i}}{t_{f}-t_{i}} [/latex] (2.6).

[latex] \bar{a}_{x}=\frac{v_{x f}-v_{x i}}{t} \approx \frac{30 m / s -10 m / s }{10 s } [/latex].

[latex] =2 m / s ^{2} [/latex].

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. Do not be afraid to attempt making educated guesses and doing some fairly drastic number rounding to simplify estimations.
Physicists engage in this type of thought analysis all the time.

(B) How far did you go during the first half of the time interval during which you accelerated?

Let us assume that the acceleration is constant, with the value calculated in part (A). Because the motion takes place in a straight line and the velocity is always in the same direction, the distance traveled from the starting point is equal to the final position of the car. We can calculate the final position at 5 s from Equation 2.12:

[latex] x_{f}=x_{i}+v_{x i} t+\frac{1}{2} a_{x} t^{2} \quad ext { (for constant } a_{x} ext { ) } [/latex] (2.12).

[latex] x_{f}=x_{i}+v_{x i} t+\frac{1}{2} a_{x} t^{2} [/latex].

[latex] \approx 0+(10 m / s )(5 s )+\frac{1}{2}\left(2 m / s ^{2} ight)(5 s )^{2}=50 m +25 m [/latex].

= 75 m.

This result indicates that if you had not accelerated, your initial velocity of 10 m/s would have resulted in a 50-m movement up the ramp during the first 5 s. The additional 25 m is the result of your increasing velocity during that interval.

Question 2.6: (A) Estimate your average acceleration as you drive up the e...

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