A truck moves a distance of 50 km. It covers first half of the distance at speed of 200 m/s and second half at speed v. If average speed of truck is 100 m/s then value of v is

Options

A. 200 m/s
B. `(200)/3 m/s`
C. `(100)/3 m/s`
D. 100 m/s

Answer

To solve the problem step by step, we will follow these logical steps:

### Step 1: Understand the Problem
The truck covers a total distance of 50 km. It travels the first half (25 km) at a speed of 200 m/s and the second half (25 km) at an unknown speed \( v \). We need to find the value of \( v \) given that the average speed for the entire journey is 100 m/s.

### Step 2: Convert Distance to Meters
Since the speeds are given in meters per second, we need to convert the distance from kilometers to meters:
\[ \text{Total distance} = 50 \text{ km} = 50 \times 1000 = 50000 \text{ m} \]
Thus, each half of the distance is:
\[ \text{Half distance} = \frac{50000}{2} = 25000 \text{ m} \]

### Step 3: Calculate Time for the First Half
Using the formula for time, \( t = \frac{d}{v} \):
\[ t_1 = \frac{25000 \text{ m}}{200 \text{ m/s}} = 125 \text{ s} \]

### Step 4: Calculate Total Time for the Journey
The average speed is given as 100 m/s. The formula for average speed is:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \]
Rearranging gives us:
\[ \text{Total time} = \frac{\text{Total distance}}{\text{Average speed}} = \frac{50000 \text{ m}}{100 \text{ m/s}} = 500 \text{ s} \]

### Step 5: Calculate Time for the Second Half
Let \( t_2 \) be the time taken to cover the second half of the distance. The total time for the journey is the sum of the times for both halves:
\[ t = t_1 + t_2 \]
Substituting the known values:
\[ 500 \text{ s} = 125 \text{ s} + t_2 \]
Solving for \( t_2 \):
\[ t_2 = 500 \text{ s} - 125 \text{ s} = 375 \text{ s} \]

### Step 6: Calculate the Speed for the Second Half
Now, we can find the speed \( v \) for the second half using the formula \( v = \frac{d}{t} \):
\[ v = \frac{25000 \text{ m}}{375 \text{ s}} = \frac{25000}{375} \text{ m/s} \]
Simplifying this:
\[ v = \frac{25000 \div 125}{375 \div 125} = \frac{200}{3} \text{ m/s} \]

### Final Answer
Thus, the value of \( v \) is:
\[ v = \frac{200}{3} \text{ m/s} \]

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