The velocity of train increases uniformly from 20 km/h to 60 km/h in 4 hour. The distance travelled by the train during this period is

Options

A. 160 km
B. 180 km
C. 100 km
D. 120 km

Answer

To find the distance traveled by the train as its velocity increases uniformly from 20 km/h to 60 km/h over a period of 4 hours, we can follow these steps:

### Step 1: Determine the initial and final velocities
- Initial velocity (u) = 20 km/h
- Final velocity (v) = 60 km/h

### Step 2: Calculate the acceleration
Acceleration (a) can be calculated using the formula:
\[ a = \frac{v - u}{t} \]
where:
- \( v \) = final velocity
- \( u \) = initial velocity
- \( t \) = time in hours

Substituting the values:
\[ a = \frac{60 \text{ km/h} - 20 \text{ km/h}}{4 \text{ h}} = \frac{40 \text{ km/h}}{4 \text{ h}} = 10 \text{ km/h}^2 \]

### Step 3: Use the formula for distance traveled under uniform acceleration
The distance (s) traveled can be calculated using the formula:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( u \) = initial velocity
- \( a \) = acceleration
- \( t \) = time

Substituting the known values:
\[ s = (20 \text{ km/h}) \times (4 \text{ h}) + \frac{1}{2} \times (10 \text{ km/h}^2) \times (4 \text{ h})^2 \]

### Step 4: Calculate the first term
Calculating the first term:
\[ 20 \text{ km/h} \times 4 \text{ h} = 80 \text{ km} \]

### Step 5: Calculate the second term
Calculating the second term:
\[ \frac{1}{2} \times 10 \text{ km/h}^2 \times 16 \text{ h}^2 = 5 \text{ km/h}^2 \times 16 \text{ h}^2 = 80 \text{ km} \]

### Step 6: Add both terms to find the total distance
Now, add both terms:
\[ s = 80 \text{ km} + 80 \text{ km} = 160 \text{ km} \]

### Final Answer
The distance traveled by the train during this period is **160 km**.

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