For a train engine moving with speed of \(20\, ms ^{-1}\), the driver must apply brakes at a distance of 500 \(m\)before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed \(\sqrt{x} ms ^{-1}\). The value of \(x\) is ________ . (Assuming same retardation is produced by brakes)
For a train engine moving with speed of \(20\, ms ^{-1}\), the driver must apply brakes at a distance of 500 \(m\)before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed \(\sqrt{x} ms ^{-1}\). The value of \(x\) is ________ . (Assuming same retardation is produced by brakes)
Show Hint
Remember the equations of motion and apply them carefully, paying attention to the signs of the quantities involved.
Correct Answer: 200
Solution and Explanation
Step 1: Calculate the Retardation
Given initial velocity \( u = 20 \, \text{m/s} \), distance \( S_1 = 500 \, \text{m} \), and final velocity \( v = 0 \). Using the third equation of motion:
\[ v^2 = u^2 + 2aS \] \[ 0 = (20)^2 + 2a(500) \] \[ 0 = 400 + 1000a \] \[ a = -\frac{400}{1000} = -0.4 \, \text{m/s}^2 \]
The negative sign indicates retardation.
Step 2: Calculate the Velocity at Half the Distance
Now, the brakes are applied at half the distance, so \( S_2 = \frac{500}{2} = 250 \, \text{m} \). The initial velocity is still \( u = 20 \, \text{m/s} \). We need to find the final velocity (\( v \)) when the train crosses the station. Using the third equation of motion:
\[ v^2 = u^2 + 2aS_2 \] \[ v^2 = (20)^2 + 2(-0.4)(250) \] \[ v^2 = 400 - 200 \] \[ v^2 = 200 \] \[ v = \sqrt{200} \, \text{m/s} \]
Step 3: Find the Value of \( x \)
The velocity is given as \( \sqrt{x} \, \text{m/s} \). We have found that \( v = \sqrt{200} \, \text{m/s} \). Therefore,
\[ x = 200 \]
Conclusion: The value of \( x \) is 200.
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Concepts Used:
Motion in a straight line
The motion in a straight line is an object changes its position with respect to its surroundings with time, then it is called in motion. It is a change in the position of an object over time. It is nothing but linear motion.
Types of Linear Motion:
Linear motion is also known as the Rectilinear Motion which are of two types:
- Uniform linear motion with constant velocity or zero acceleration: If a body travels in a straight line by covering an equal amount of distance in an equal interval of time then it is said to have uniform motion.
- Non-Uniform linear motion with variable velocity or non-zero acceleration: Not like the uniform acceleration, the body is said to have a non-uniform motion when the velocity of a body changes by unequal amounts in equal intervals of time. The rate of change of its velocity changes at different points of time during its movement.