Changing Units — Question 4

Question 4

Per second and per hour are not interchangeable labels

A cart’s speed increases as shown. In this model acceleration is constant and is defined by $a=\Delta v/\Delta t$.

Use exact $1\ \mathrm{km}=1000\ \mathrm{m}$ and $1\ \mathrm{h}=3600\ \mathrm{s}$. Treat the given speeds and times as exact.

1. Calculate $a$ in $\mathrm{m/s^2}$ by converting the speed change before dividing by elapsed time.

2. A display reports $9.0\ (\mathrm{km/h})/\mathrm{s}$. Is this consistent with your answer? Convert its complete unit expression.

3. A second display reports $9.0\ \mathrm{km/h^2}$. Convert this to $\mathrm{m/s^2}$ and compare it with the first display. Explain why the two 9.0 readings do not describe the same acceleration.

4. Give the correct numerical acceleration in $\mathrm{km/h^2}$ for the cart. Use the exponent on hour explicitly rather than guessing a power of 3600.

Question 4 - Solution

Per second and per hour are not interchangeable labels

(a) Convert the speed change first.

In the first $2.0\ \mathrm{s}$ the speed changes by $18.0(1000/3600)\ \mathrm{m/s}=5.0\ \mathrm{m/s}$. Therefore $$a=\frac{5.0\ \mathrm{m/s}}{2.0\ \mathrm{s}}=\boxed{2.5\ \mathrm{m/s^2}}.$$ The next interval gives the same result.

(b) Preserve the mixed denominator.

$$9.0\ \frac{\mathrm{km}}{\mathrm{h\,s}} \frac{1000\ \mathrm{m}}{1\ \mathrm{km}} \frac{1\ \mathrm{h}}{3600\ \mathrm{s}} =2.5\ \mathrm{m/s^2}.$$ The first display is consistent. It describes a change of $9.0\ \mathrm{km/h}$ in each second.

(c) An hour in each denominator factor.

$$\frac{9\ \mathrm{km\,h^{-1}\,s^{-1}}}{9\ \mathrm{km\,h^{-2}}}=\frac{1\ \mathrm h}{1\ \mathrm s}=3600.$$

$$9.0\ \frac{\mathrm{km}}{\mathrm{h^2}} \frac{1000\ \mathrm{m}}{1\ \mathrm{km}} \left(\frac{1\ \mathrm{h}}{3600\ \mathrm{s}}\right)^2 =\frac{1}{1440}\ \mathrm{m/s^2}\approx0.00069444\ \mathrm{m/s^2}.$$ This is 3600 times smaller than the correct acceleration. It describes a change of $9.0\ \mathrm{km/h}$ per hour rather than per second.

(d) Convert the full squared time unit.

$$2.5\ \frac{\mathrm{m}}{\mathrm{s^2}} \frac{1\ \mathrm{km}}{1000\ \mathrm{m}} \left(\frac{3600\ \mathrm{s}}{1\ \mathrm{h}}\right)^2 =\boxed{32\,400\ \mathrm{km/h^2}}.$$ The same value follows from $9.0\ (\mathrm{km/h})/\mathrm{s}$ multiplied by $3600\ \mathrm{s/h}$.