Two cars are moving with speed `v_(1), v_(2)` towards a crossing along two roads. If their distance from the crossing be 40 metres and 50 metres at an instant of time then they do not collide if their speed are such that

Options

A. `v_(1): v_(2)= 16:25`
B. `v_(1): v_(2) ne 4:5`
C. `v_(1): v_(2) ne 5 : 4`
D. `v_(1): v_(2)= 25: 16`

Answer

To determine the conditions under which two cars moving towards a crossing do not collide, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( v_1 \) = speed of Car A
- \( v_2 \) = speed of Car B
- \( d_1 = 40 \) meters (distance of Car A from the crossing)
- \( d_2 = 50 \) meters (distance of Car B from the crossing)

### Step 2: Calculate the Time Taken for Each Car to Reach the Crossing
The time taken for each car to reach the crossing can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Thus, for Car A:
\[ T_1 = \frac{d_1}{v_1} = \frac{40}{v_1} \]
And for Car B:
\[ T_2 = \frac{d_2}{v_2} = \frac{50}{v_2} \]

### Step 3: Set Up the Condition for No Collision
For the two cars to not collide, they must not reach the crossing at the same time. This means:
\[ T_1 \neq T_2 \]
Substituting the expressions for \( T_1 \) and \( T_2 \):
\[ \frac{40}{v_1} \neq \frac{50}{v_2} \]

### Step 4: Cross Multiply to Eliminate the Fractions
Cross multiplying gives us:
\[ 40 \cdot v_2 \neq 50 \cdot v_1 \]

### Step 5: Rearranging the Inequality
We can rearrange this to express the relationship between the speeds:
\[ \frac{v_1}{v_2} \neq \frac{40}{50} \]
This simplifies to:
\[ \frac{v_1}{v_2} \neq \frac{4}{5} \]

### Conclusion
Thus, the two cars do not collide if their speeds are such that:
\[ \frac{v_1}{v_2} \neq \frac{4}{5} \]