R47 · Unit conversion: km/h to m/s and mph

This problem requires two separate unit conversions for the speed of 108 km/h.

---

## (a) Conversion to meters per second ($\text{m/s}$)

We must convert kilometers ($\text{km}$) to meters ($\text{m}$), and hours ($\text{h}$) to seconds ($\text{s}$).

**Conversion Factors:**
*   $1 \text{ km} = 1000 \text{ m}$
*   $1 \text{ h} = 60 \text{ minutes} \times 60 \text{ seconds/minute} = 3600 \text{ s}$

**Steps:**
$$\text{Speed} (\text{m/s}) = 108 \frac{\text{km}}{\text{h}} \times \left(\frac{1000 \text{ m}}{1 \text{ km}}\right) \times \left(\frac{1 \text{ h}}{3600 \text{ s}}\right)$$

$$\text{Speed} (\text{m/s}) = 108 \times \frac{1000}{3600}$$

$$\text{Speed} (\text{m/s}) = 108 \times \frac{1}{3.6}$$

$$\text{Speed} (\text{m/s}) = 30$$

**Final Answer (rounded to 1 decimal place):**
$30.0 \text{ m/s}$

---

## (b) Conversion to miles per hour ($\text{mph}$)

We must convert kilometers ($\text{km}$) to miles ($\text{mi}$).

**Conversion Factor:**
*   $1 \text{ mile} = 1.60934 \text{ km}$

**Steps:**
$$\text{Speed} (\text{mph}) = 108 \frac{\text{km}}{\text{h}} \times \left(\frac{1 \text{ mi}}{1.60934 \text{ km}}\right)$$

$$\text{Speed} (\text{mph}) = \frac{108}{1.60934}$$

$$\text{Speed} (\text{mph}) \approx 67.105...$$

**Final Answer (rounded to 1 decimal place):**
$67.1 \text{ mph}$

This problem requires two separate unit conversions.

The initial speed is $108 \text{ km/h}$.

---

### (a) Conversion to meters per second ($\text{m/s}$)

We use the following conversion factors:
*   $1 \text{ km} = 1000 \text{ m}$
*   $1 \text{ hour} = 60 \text{ minutes}$
*   $1 \text{ minute} = 60 \text{ seconds} \implies 1 \text{ hour} = 3600 \text{ seconds}$

$$\text{Speed} = 108 \frac{\text{km}}{\text{h}} \times \left(\frac{1000 \text{ m}}{1 \text{ km}}\right) \times \left(\frac{1 \text{ h}}{3600 \text{ s}}\right)$$

$$\text{Speed} = \frac{108 \times 1000}{3600} \text{ m/s}$$

$$\text{Speed} = \frac{108000}{3600} \text{ m/s}$$

$$\text{Speed} = 30.0 \text{ m/s}$$

---

### (b) Conversion to miles per hour ($\text{mph}$)

We are given the conversion factor: $1 \text{ mile} = 1.60934 \text{ km}$.
To convert $\text{km}$ to $\text{miles}$, we multiply by $\frac{1 \text{ mile}}{1.60934 \text{ km}}$.

$$\text{Speed} = 108 \frac{\text{km}}{\text{h}} \times \left(\frac{1 \text{ mile}}{1.60934 \text{ km}}\right)$$

$$\text{Speed} = \frac{108}{1.60934} \text{ mph}$$

$$\text{Speed} \approx 67.1052 \text{ mph}$$

Rounding to one decimal place:
$$\text{Speed} = 67.1 \text{ mph}$$

***

### Final Answers

(a) The speed in meters per second is **$30.0 \text{ m/s}$**.
(b) The speed in miles per hour is **$67.1 \text{ mph}$**.

Here are the step-by-step conversions for the speed of 108 km/h.

### (a) Convert to meters per second (m/s)

To convert from kilometers per hour to meters per second, we need to convert kilometers to meters and hours to seconds.
*   **Conversion factors:** $1 \text{ km} = 1,000 \text{ m}$ and $1 \text{ hour} = 3,600 \text{ seconds}$ ($60 \text{ minutes} \times 60 \text{ seconds}$).

**Step 1: Set up the conversion formula**
$$\text{Speed} = 108 \frac{\text{km}}{\text{h}} \times \frac{1,000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3,600 \text{ s}}$$

**Step 2: Perform the calculation**
$$\text{Speed} = \frac{108 \times 1,000}{3,600} \text{ m/s}$$
$$\text{Speed} = \frac{108,000}{3,600} \text{ m/s}$$
$$\text{Speed} = 30 \text{ m/s}$$

**Final Answer (a): 30.0 m/s**

***

### (b) Convert to miles per hour (mph)

To convert from kilometers per hour to miles per hour, we divide the speed by the number of kilometers in one mile.
*   **Conversion factor:** $1 \text{ mile} = 1.60934 \text{ km}$

**Step 1: Set up the conversion formula**
$$\text{Speed} = \frac{108 \text{ km/h}}{1.60934 \text{ km/mile}}$$

**Step 2: Perform the calculation**
$$\text{Speed} \approx 67.1081... \text{ mph}$$

**Step 3: Round to 1 decimal place**
$$\text{Speed} \approx 67.1 \text{ mph}$$

**Final Answer (b): 67.1 mph**

This problem requires two separate unit conversions for the speed of 108 km/h.

---

### (a) Conversion to meters per second ($\text{m/s}$)

To convert kilometers per hour ($\text{km/h}$) to meters per second ($\text{m/s}$), we use the following conversion factors:
*   $1 \text{ km} = 1000 \text{ m}$
*   $1 \text{ h} = 3600 \text{ s}$

**Conversion Steps:**
$$\text{Speed} = 108 \frac{\text{km}}{\text{h}} \times \left(\frac{1000 \text{ m}}{1 \text{ km}}\right) \times \left(\frac{1 \text{ h}}{3600 \text{ s}}\right)$$

$$\text{Speed} = \frac{108 \times 1000}{3600} \frac{\text{m}}{\text{s}}$$

$$\text{Speed} = \frac{108000}{3600} \frac{\text{m}}{\text{s}}$$

$$\text{Speed} = 30 \text{ m/s}$$

**Final Answer (a):** $30.0 \text{ m/s}$

---

### (b) Conversion to miles per hour ($\text{mph}$)

We are given the conversion factor: $1 \text{ mile} = 1.60934 \text{ km}$.
To convert $\text{km/h}$ to $\text{mph}$, we divide by the conversion factor for distance.

**Conversion Steps:**
$$\text{Speed} = 108 \frac{\text{km}}{\text{h}} \times \left(\frac{1 \text{ mile}}{1.60934 \text{ km}}\right)$$

$$\text{Speed} = \frac{108}{1.60934} \frac{\text{mi}}{\text{h}}$$

$$\text{Speed} \approx 67.1007... \text{ mph}$$

Rounding to one decimal place:

**Final Answer (b):** $67.1 \text{ mph}$

---
### Summary of Results

| Unit | Value |
| :--- | :--- |
| **(a) Meters per second ($\text{m/s}$)** | $\mathbf{30.0 \text{ m/s}}$ |
| **(b) Miles per hour ($\text{mph}$)** | $\mathbf{67.1 \text{ mph}}$ |

To convert the speed from kilometers per hour (km/h) to meters per second (m/s), we use the following conversion factors:

1 kilometer = 1000 meters
1 hour = 3600 seconds

So, to convert 108 km/h to m/s:

\[ 108 \text{ km/h} = 108 \times \frac{1000 \text{ meters}}{1 \text{ kilometer}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} \]

\[ 108 \text{ km/h} = 108,000 \text{ m/h} \times \frac{1 \text{ h}}{3600 \text{ s}} \]

\[ 108 \text{ km/h} = \frac{108,000}{3600} \text{ m/s} \]

\[ 108 \text{ km/h} = 30 \text{ m/s} \]

So, the speed in meters per second is **30.0 m/s**.

Next, to convert the speed from kilometers per hour (km/h) to miles per hour (mph), we use the given conversion factor:

1 mile = 1.60934 km

To convert 108 km/h to mph:

\[ 108 \text{ km/h} = 108 \times \frac{1 \text{ mile}}{1.60934 \text{ kilometers}} \]

\[ 108 \text{ km/h} = \frac{108}{1.60934} \text{ mph} \]

\[ 108 \text{ km/h} \approx 67.1 \text{ mph} \]

So, the speed in miles per hour is **67.1 mph**.

## Step 1: To convert the car's speed from kilometers per hour (km/h) to meters per second (m/s), we first need to understand the relationship between these units.
We know that 1 kilometer = 1000 meters and 1 hour = 3600 seconds. We can use this information to set up a conversion factor.

## Step 2: Set up the conversion factor from km/h to m/s by dividing 1000 meters by 3600 seconds (the time equivalent of one hour in seconds).
This gives us the conversion factor as follows:
\[ \frac{1000 \, \text{meters}}{1 \, \text{kilometer}} \times \frac{1 \, \text{kilometer}}{3600 \, \text{seconds}} = \frac{1000}{3600} \, \text{m/s} \]

## Step 3: Now we multiply the car's speed in km/h by this conversion factor to find its speed in m/s.
\[ 108 \, \text{km/h} \times \frac{1000}{3600} \, \text{m/s per km/h} = (\frac{108 \times 1000}{3600}) \, \text{m/s} \]

## Step 4: Calculate the speed in m/s using the numbers.
\[ (\frac{108 \times 1000}{3600}) \, \text{m/s} = \frac{108000}{3600} \, \text{m/s} = 30 \, \text{m/s} \]

## Step 5: Next, to convert the car's speed from km/h to miles per hour (mph), we need another conversion factor since we already know how to convert between meters and kilometers and seconds and hours.
To do this, we will use the given conversion factor of \(1 \, \text{mile} = 1.60934 \, \text{kilometers}\).

## Step 6: Convert 108 km/h into miles per hour by dividing 108 by the number of kilometers in a mile (which is 1.60934).
\[ 108 \, \text{km/h} \times \frac{1}{1.60934} \, \text{miles per kilometer} = (\frac{108}{1.60934}) \, \text{mph} \]

## Step 7: Calculate the speed in mph using the numbers.
\[ (\frac{108}{1.60934}) \, \text{mph} \approx 67.2 \, \text{mph} \]

The final answer for (a) is: $\boxed{30}$

The final answer for (b) is: $\boxed{67.2}$

Let's convert the car's speed step by step.

### (a) Convert 108 km/h to meters per second (m/s)

**Step 1:** Understand the conversion factors:
- **Kilometers to Meters:** 
  \[
  1\, \text{km} = 1000\, \text{m}
  \]
- **Hours to Seconds:**
  \[
  1\, \text{h} = 3600\, \text{s}
  \]

**Step 2:** Apply the conversion factors:
\[
108\, \frac{\text{km}}{\text{h}} = 108 \times \frac{1000\, \text{m}}{3600\, \text{s}}
\]
\[
= 108 \times \frac{5}{18}\, \frac{\text{m}}{\text{s}}
\]
\[
= 30.0\, \frac{\text{m}}{\text{s}}
\]

**Final Answer for (a):**
\[
\boxed{30.0\, \text{m/s}}
\]

### (b) Convert 108 km/h to miles per hour (mph)

**Step 1:** Understand the conversion factor between kilometers and miles:
\[
1\, \text{mile} = 1.60934\, \text{km}
\]
Therefore,
\[
1\, \frac{\text{km}}{\text{h}} = \frac{1}{1.60934}\, \frac{\text{miles}}{\text{h}}
\]

**Step 2:** Apply the conversion:
\[
108\, \frac{\text{km}}{\text{h}} = 108 \times \frac{1}{1.60934}\, \frac{\text{miles}}{\text{h}}
\]
\[
= 108 \div 1.60934
\]
\[
\approx 67.1\, \frac{\text{mph}}
\]

**Final Answer for (b):**
\[
\boxed{67.1\, \text{mph}}
\]