Visit Motion 1 for a quick review of the formulas used in motion problems and some examples.
You just bought your first car and are excited to drive it. However, the ad mentioned that the speedometer doesn't work properly and needs to be replaced.
You aren't too worried and plan to drive cautiously until you have some money to replace it. Just as you pull into school on your first drive, a police car turns on its sirens and hands you your first ticket for speeding.
Confused, you think about your drive into school: The speed limit is \(60 \; [\text{km/hr}]\) and your speedometer also read \(60 \; [\text{km/hr}]\). It took \(10\) minutes to drive to school.
On the way home, to play it safe, you drive \(50 \; [\text{km/hr}]\) and it takes \(14\) minutes to get home after dropping your friend off. At the end of the day, your odometer reads \(25.8 \; [\text{km}]\). How fast were you driving when you got the ticket?
In this question, we are asked for the speed, \(v\) on the way to school when you received a ticket.
Let's set up a table to organize the information we know. Let \(d\) represent the distance travelled in \([\text{km}]\). Let \(t\) represent the time in \([\text{hr}]\). Let \(v\) represent the speed on the way to school in \([\text{km/hr}]\):
| Trip | \(d\) | \(v\) | \(t\) |
|---|---|---|---|
| School | \(d_1\) | \(v\) | \(\cfrac{10}{60}\) |
| Home | \(d_2\) | \(v-10\) | \(\cfrac{14}{60}\) |
Notice that the speed on the way home is \(10\) less than on the way to school. Also, don't forget to convert the time to hours.
Unfortunately, we do not know the distance travelled to school or on the way home. But, we know the total distance travelled is \(25.8\;[\text{km}]\) from the odometer.
We can write this as:
Remember the formula \( d = vt \). We can use this for \( d_1 \) and \( d_2 \):
| School | \(d_1 = \cfrac{10}{60}v \) |
| Home | \(d_2 = \cfrac{14}{60}(v-10) \) |
We can use substitution to get one equation with one unknown. We can then simplify this equation to determine the speed you were driving when you got your ticket:
\(d_1 + d_2 = 25.8\)
\(\cfrac{10}{60}v + \cfrac{14}{60}(v-10) = 25.8\)
\(\cfrac{10}{60}v + \cfrac{14}{60}v - \cfrac{140}{60} = 25.8\)
\(v \left(\cfrac{10}{60} + \cfrac{14}{60}\right) - \cfrac{140}{60} = 25.8\)
\(v \left(\cfrac{10}{60} + \cfrac{14}{60}\right) = 25.8 + \cfrac{140}{60}\)
\(v \left(\cfrac{24}{60}\right) = 28.13\)
\(v = \cfrac{28.13}{0.4} \)
\(v \approx 70 \; \left[\cfrac{\text{km}}{\text{hr}}\right]\)
Looks like you were driving a little fast on the way to school! On average, you drove \(\boldsymbol{70 \left[\cfrac{\textbf{km}}{\textbf{hr}}\right]}\) when the limit was \(60 \; [\text{km/hr}]\).