An express train moving at 30 m/s reduces its speed to 10 m/s in a distance of 240 m. If the breaking force is increased by 12.5% in the beginning find the distance that it travels before coming to rest 

Options

A. 270 m
B. 240 m
C. 210 m
D. 195 m

Answer

To solve the problem step by step, we will use the equations of motion and the concept of acceleration due to braking force.

### Step 1: Identify the given values
- Initial velocity (u) = 30 m/s
- Final velocity (v) = 10 m/s
- Distance traveled (s) = 240 m

### Step 2: Use the first equation of motion
We will use the equation:
\[ v^2 - u^2 = 2as \]
where \( a \) is the acceleration (which will be negative since it is deceleration).

Substituting the known values:
\[ 10^2 - 30^2 = 2a(240) \]
\[ 100 - 900 = 480a \]
\[ -800 = 480a \]
\[ a = \frac{-800}{480} = -\frac{5}{3} \text{ m/s}^2 \]

### Step 3: Calculate the new acceleration after increasing the braking force
The braking force is increased by 12.5%. The new acceleration \( a' \) can be calculated as:
\[ a' = 1.125 \times a \]
Substituting the value of \( a \):
\[ a' = 1.125 \times \left(-\frac{5}{3}\right) \]
\[ a' = -\frac{5 \times 1.125}{3} = -\frac{5.625}{3} = -1.875 \text{ m/s}^2 \]

### Step 4: Use the second equation of motion to find the distance to come to rest
Now, we need to find the distance \( s' \) that the train travels before coming to rest (final velocity \( v = 0 \)):
Using the equation:
\[ v^2 - u^2 = 2a's' \]
Substituting \( v = 0 \), \( u = 10 \) m/s (the speed at which we are starting the new deceleration), and \( a' = -1.875 \):
\[ 0^2 - 10^2 = 2(-1.875)s' \]
\[ -100 = -3.75s' \]
\[ s' = \frac{100}{3.75} = 26.67 \text{ m} \]

### Step 5: Calculate the total distance traveled
The total distance traveled before coming to rest is the distance already traveled (240 m) plus the distance traveled during the new deceleration (26.67 m):
\[ \text{Total distance} = 240 + 26.67 = 266.67 \text{ m} \]

### Final Answer
The total distance that the train travels before coming to rest is approximately **266.67 meters**.

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