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):**
\[
\boxed{67.1\, \text{mph}}
\]

Here are your conversions:

(a) To convert kilometers per hour to meters per second, we use the formula: 

$$
\begin{equation*}
    \text{km/h} \cdot \frac{1000~m}{1 ~ km } \cdot \frac{1 ~ h}{3600 ~ s}\,.
\end{equation*}
$$

Plugging in the given value, we get: 

$$
\begin{align*}
    v_\text{(mps)} &= 108~ \frac{\text{km}}{\text{h}} \cdot \frac{1000 ~ m}{1 ~ \text{km}} \cdot \frac{1 ~ \text{h}}{3600 ~ \text{s}}\\
    &= 30 ~\frac{\text{m}}{\text{s}}
\end{align*}
$$ 

so the speed in meters per second is $30.0~ \frac{\text{m}}{\text{s}}$.

(b) To convert kilometers per hour to miles per hour, we use the given conversion factor: 

$$
\begin{equation*}
    1 ~ \text{mile} = 1.60934 ~ \text{km}\,.
\end{equation*}
$$ 

Then: 

$$
\begin{align*}
    v_\text{(mph)} &= 108~ \frac{\text{km}}{\text{h}} \cdot \frac{1 ~ \text{mile}}{1.60934 ~ \text{km}}\\
    &\approx 67.2 ~\frac{\text{miles}}{\text{h}}
\end{align*}
$$ 

So the speed in miles per hour is approximately $67.2~ \frac{\text{miles}}{\text{h}}$.

<think>
Okay, so I need to convert the car's speed from kilometers per hour to meters per second and also to miles per hour. Let's see... The car is going 108 km/h. 

Starting with part (a): converting to meters per second. Hmm, right, because kilometers to meters and hours to seconds. I remember that 1 kilometer is 1000 meters, so maybe I can multiply by that to get meters. And then hours to seconds... there are 60 minutes in an hour and 60 seconds in a minute, so 60 times 60 is 3600 seconds in an hour. So maybe I can divide by 3600 to get seconds? Let me write that out step by step.

First, 108 km/h. To convert km to m, multiply by 1000, because 1 km = 1000 m. So 108 km/h * 1000 m/km. That gives me meters per hour. Then to convert hours to seconds, I need to divide by the number of seconds in an hour, which is 3600. So, 108 * 1000 divided by 3600. Let me compute that.

Calculating numerator first: 108 * 1000 is 108,000. Then divide by 3600. Hmm, 108,000 divided by 3600. Maybe simplify the fraction? Let me see. 108,000 divided by 3600. Let me divide both numerator and denominator by 100 first to make it simpler? Wait, 108,000 divided by 3600. Let me see, 3600 is 36*100, and 108,000 is 108*1000. Maybe factor out 100? Let me see, 108,000 / 3600 = (108 / 36) * (1000 / 100). Because 108,000 is 108*1000 and 3600 is 36*100. So 108 divided by 36 is 3, and 1000 divided by 100 is 10. So 3 * 10 is 30. Wait, so that would be 30? Wait, that seems too easy. Let me check again.

Alternatively, let me do 108,000 divided by 3600. Let me divide both numerator and denominator by 100: 1080 divided by 36. 1080 divided by 36... 36 times 30 is 1080. So that's 30. Hmm, okay, so that gives me 30 meters per second? That seems reasonable. Let me confirm with another approach. 

Alternatively, if the car is going 108 km/h, so in one hour, it goes 108 km. To find meters per second, since 1 hour is 3600 seconds, the distance in meters is 108,000 meters. So 108,000 meters divided by 3600 seconds equals 30 m/s. Yeah, that's the same answer. Okay, so part (a) is 30.0 m/s? Wait, the question says to round to one decimal place. 30.0 is already to one decimal. So that's okay.

Now part (b): converting 108 km/h to miles per hour. They gave the conversion factor 1 mile is 1.60934 km. So since we have kilometers, we need to convert to miles. Since 1 mile is 1.60934 km, then 1 km is 1/1.60934 miles. So to convert km to miles, divide by 1.60934. 

So the speed in miles per hour would be 108 km/h divided by 1.60934 km/mile. Let me compute that. 

So 108 divided by 1.60934. Let me do this division. Maybe approximate first? Since 1 mile is approximately 1.6 km, so 108 divided by 1.6 would be roughly 67.5, but since the actual conversion factor is a bit more than 1.6, the actual result should be a bit less than 67.5. Let me compute exactly:

108 / 1.60934. Let me use calculator steps.

Alternatively, I can write it as 108 ÷ 1.60934.

Let me approximate this. Let me see, 1.60934 times 67 is let's calculate that:

1.60934 * 60 = 96.5604

1.60934 * 7 = 11.26538

Adding those gives 96.5604 + 11.26538 = 107.82578

So 1.60934 * 67 ≈ 107.82578, which is very close to 108. So 67 would give us approximately 107.825, so actually, 67 is almost exactly 108 divided by 1.60934, but a tiny bit less. Since 67 * 1.60934 is ~107.825, so 108 would be 67 + (108 - 107.825)/1.60934.

The difference is 108 - 107.82578 = 0.17422 km.

So 0.17422 km / 1.60934 km/mile is 0.17422 /1.60934 ≈ 0.108 miles.

Therefore, total is approximately 67.108. So around 67.1 miles per hour.

Wait, but let me use a calculator-like approach more precisely.

Alternatively, let me do 108 divided by 1.60934:

Let me write this as 108 / 1.60934. Let me compute this:

First, 1.60934 x 67 = 107.82578 as above. 

So 108 - 107.82578 = 0.17422 left. 

So 0.17422 divided by 1.60934 equals approximately 0.1082. 

So adding that to 67 gives 67.1082. So approximately 67.108. 

Therefore,

**Solution:**

We are tasked with converting a car's speed from kilometers per hour (km/h) to two different units:

1. **Meters per second (m/s)**
2. **Miles per hour (mph)**

Let's go through each conversion step by step.

---

### (a) Conversion to Meters Per Second (m/s)

**Given:**  
\( 108 \, \text{km/h} \)

**Conversion Factors:**
- \( 1 \, \text{km} = 1000 \, \text{m} \)
- \( 1 \, \text{hour} = 3600 \, \text{seconds} \)

**Steps:**

1. **Convert kilometers to meters:**  
   Multiply by \( 1000 \) to convert kilometers to meters.
   \[
   108 \, \text{km/h} \times 1000 \, \frac{\text{m}}{\text{km}} = 108,000 \, \frac{\text{m}}{\text{h}}
   \]

2. **Convert hours to seconds:**  
   Divide by \( 3600 \) to convert hours to seconds.
   \[
   108,000 \, \frac{\text{m}}{\text{h}} \div 3600 \, \frac{\text{s}}{\text{h}} = 30 \, \frac{\text{m}}{\text{s}}
   \]

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

---

### (b) Conversion to Miles Per Hour (mph)

**Given:**  
\( 108 \, \text{km/h} \)

**Conversion Factor:**
- \( 1 \, \text{mile} = 1.60934 \, \text{km} \)

**Steps:**

1. **Convert kilometers to miles:**  
   Divide by the conversion factor to change kilometers to miles.
   \[
   108 \, \frac{\text{km}}{\text{h}} \div 1.60934 \, \frac{\text{km}}{\text{mile}} = 67.1 \, \frac{\text{miles}}{\text{h}}
   \]

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

---

### Summary:

- **\( 108 \, \text{km/h} = \boxed{30.0 \, \text{m/s}} \)**
- **\( 108 \, \text{km/h} = \boxed{67.1 \, \text{mph}} \)**