An object has velocity `vecv_1` w.r.t. ground. An observer moving with constant velocity `vecv_0` w.r.t. ground measures the velocity of the object as `vecv_2`. The magnitudes of three velocities are related by

Options

A. `v_0gev_1+v_2`
B. `v_1lev_2+v_0`
C. `v_2gev_1+v_0`
D. All of the above

Explanation

To solve the problem, we need to relate the velocities of the object, the observer, and the object as measured by the observer using vector addition and the triangle inequality.

### Step-by-Step Solution:

1. **Define the Velocities**:
- Let \( \vec{v_1} \) be the velocity of the object with respect to the ground.
- Let \( \vec{v_0} \) be the velocity of the observer with respect to the ground.
- Let \( \vec{v_2} \) be the velocity of the object as measured by the observer.

2. **Relative Velocity Concept**:
- The velocity of the object with respect to the observer can be expressed as:
\[ \vec{v_2} = \vec{v_1} - \vec{v_0} \]
- This equation shows that to find the velocity of the object relative to the observer, we subtract the observer's velocity from the object's velocity.

3. **Magnitude Relation**:
- The magnitudes of these velocities can be related using the triangle inequality. According to the triangle inequality, for any three vectors \( \vec{A}, \vec{B}, \vec{C} \):
\[ |\vec{A} + \vec{B}| \leq |\vec{C}| \]
- In our case, we can rearrange the equation:
\[ |\vec{v_1}| = |\vec{v_2} + \vec{v_0}| \]

4. **Applying Triangle Inequality**:
- From the triangle inequality, we know:
\[ |\vec{v_2}| + |\vec{v_0}| \geq |\vec{v_1}| \]
- This means that the sum of the magnitudes of the velocities \( |\vec{v_2}| \) and \( |\vec{v_0}| \) is greater than or equal to the magnitude of \( |\vec{v_1}| \).

5. **Conclusion**:
- Therefore, the relationship between the magnitudes of the three velocities is:
\[ |\vec{v_2}| + |\vec{v_0}| \geq |\vec{v_1}| \]

### Final Answer:
The magnitudes of the three velocities are related by:
\[ |\vec{v_2}| + |\vec{v_0}| \geq |\vec{v_1}| \]