A motorcycle, a car and a bus all have-the same kinetic energy. If equal braking forces are applied to them, they come to a halt after travelling x, y and z units of distance respectively. Which one of the following is correct ?

Options

A. `xltyltz`
B. `xgtygtz`
C. `x=y=z`
D. Relationship between x, y and z cannot be determined on the basis of given data

Answer

To solve the problem, we need to analyze the relationship between kinetic energy, braking force, and the distance traveled before coming to a halt for the motorcycle, car, and bus.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
The kinetic energy (KE) of an object in motion is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity.

2. **Given Condition**:
We are told that the motorcycle, car, and bus all have the same kinetic energy. Therefore, we can denote their kinetic energies as:
\[ KE_{motorcycle} = KE_{car} = KE_{bus} = KE \]

3. **Braking Force and Work-Energy Principle**:
When a braking force \( F \) is applied, the work done by the braking force is equal to the change in kinetic energy. The work done by the force can be expressed as:
\[ W = F \cdot d \]
where \( d \) is the distance traveled before coming to a stop.

4. **Setting Up the Equation**:
Since the kinetic energy is converted into work done by the braking force, we can write:
\[ KE = F \cdot d \]
Rearranging gives us:
\[ d = \frac{KE}{F} \]

5. **Distance for Each Vehicle**:
For each vehicle, we can express the distances traveled before stopping as:
- For the motorcycle:
\[ x = \frac{KE}{F} \]
- For the car:
\[ y = \frac{KE}{F} \]
- For the bus:
\[ z = \frac{KE}{F} \]

6. **Conclusion**:
Since all three vehicles have the same kinetic energy and are subjected to the same braking force, their distances traveled before coming to a stop can be expressed as:
\[ x = y = z \]

### Final Answer:
Thus, the correct relationship is:
\[ x = y = z \]