Question

On your cousin's wedding day you leave for the church $30.0\min$ before the ceremony is to begin, which should leave plenty of time since the church is only $17.0\mathrm{km}$ away. On the way, however, you have to make an unanticipated stop because of road construction work. As a result, your average speed for the first $15\min$ is only $12.0\mathrm{km}/\mathrm{h}$. What average speed do you need to have on the remainder of the trip to get you to the church on time?

Short answer

You need to travel at 56 km/h for the remainder of the trip.

Step-by-step solution

Calculate Distance Traveled Initially

First, we need to determine how far you travel during the first 15 minutes. Since your speed is 12 km/h, convert 15 minutes into hours by dividing by 60, which is 0.25 hours. Now, use the formula: $$\text{Distance}=\text{Speed}\times \text{Time}$$Thus, the distance is:$$12\text{km/h}\times 0.25\text{h}=3\text{km}$$

Determine Remaining Distance

Subtract the distance traveled in the first 15 minutes from the total distance to the church:$$17\text{km}-3\text{km}=14\text{km}$$So, you have 14 km left to cover.

Determine Remaining Time Available

You initially had 30 minutes in total, of which 15 minutes are already used. Therefore, the remaining time you have is:$$30\text{min}-15\text{min}=15\text{min}$$Convert this time into hours:$$\frac{15}{60}=0.25\text{hours}$$

Calculate Required Average Speed for Remaining Distance

Now, find out what speed you need to travel the remaining 14 km in 0.25 hours using the speed formula rearranged:$$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$$Plug in the values:$$\text{Speed}=\frac{14\text{km}}{0.25\text{hours}}=56\text{km/h}$$

Key concepts

Distance Calculations

When solving problems related to travel and movement, understanding how to calculate distances is essential. In our example, you need to find out how far you have traveled during the first part of your journey. This involves using the formula for distance:

• Distance = Speed × Time

Here, speed is given as 12 km/h, and the time is initially 15 minutes. It's important to convert the time into hours (since speed is in km/h) by dividing the minutes by 60.
This conversion gives us 0.25 hours. By using the distance formula, we multiply the speed (12 km/h) by the time (0.25 hours) to find that the distance is 3 km.
This step ensures you have determined how much of the 17 km journey is covered during the first segment of your trip.
After calculating how far you've traveled, the next task is to find out how much distance is left. Subtract the initial distance traveled from the total distance:

• Remaining Distance = Total Distance - Distance Traveled Initially

This calculation reveals that you have 14 km remaining to travel.

Speed and Velocity

Speed and velocity are fundamental concepts in motion-related problems. Speed refers to how fast you are moving and is calculated as the distance traveled over time. In your journey to the church, the initial speed is given as 12 km/h.
But you need a new speed to cover a remaining 14 km in a certain time if you want to arrive on time. To find the required speed for the rest of your journey, use the speed formula rearranged:

• Speed = Distance / Time

Consider the new conditions: You have a remaining distance of 14 km and a time of 0.25 hours to cover it. Plug these values into the formula:
The calculated speed is 56 km/h.
This means you'll need to adjust your rate of travel significantly to meet your schedule.

Time Conversion

Time conversion is a crucial skill in physics problems when working with different units of measurement. In our journey problem, time is initially given in minutes, while speed is in kilometers per hour. This requires a conversion step.

• Minutes to Hours: since there are 60 minutes in an hour, divide the number of minutes by 60.

For instance, your travel initially takes 15 minutes, which becomes 0.25 hours when you divide by 60.
It's essential to convert time appropriately because it ensures consistency across calculations, especially when dealing with speed in km/h.
Failure to convert time units correctly can lead to incorrect calculations and misleading results. With the remaining 15 minutes of time being another 0.25 hours, this conversion aids in determining the new critical status of your pace needed to reach the destination timely.