two tains one travelling at `72kmh^(-1)` and other at `90 kmh ^(-1)` are heading towards one another along a strauight level track. When they are 1.0km apart, both the drivers simultanously see the other 's train and apply brakes which restard each train at the rate of `1 .0ms^(-2)`.determine whether the trains would collide or not.

Answer

To determine whether the two trains will collide, we can follow these steps:

### Step 1: Convert Speeds from km/h to m/s
The speeds of the trains need to be converted from kilometers per hour to meters per second.

- Train 1 speed: \( 72 \, \text{km/h} = 72 \times \frac{1000}{3600} = 20 \, \text{m/s} \)
- Train 2 speed: \( 90 \, \text{km/h} = 90 \times \frac{1000}{3600} = 25 \, \text{m/s} \)

### Step 2: Calculate the Initial Relative Velocity
Since the trains are moving towards each other, we can add their speeds to find the initial relative velocity.

\[ U_{\text{relative}} = V_1 + V_2 = 20 \, \text{m/s} + 25 \, \text{m/s} = 45 \, \text{m/s} \]

### Step 3: Determine the Total Retardation
Both trains apply brakes with a retardation of \( 1 \, \text{m/s}^2 \). The total retardation when considering both trains is:

\[ A_{\text{relative}} = A_1 + A_2 = 1 \, \text{m/s}^2 + 1 \, \text{m/s}^2 = 2 \, \text{m/s}^2 \]

### Step 4: Use the Equation of Motion
We can use the equation of motion to find out how far the trains will travel before they come to a stop:

\[ V^2 = U^2 + 2AS \]

Here, \( V = 0 \) (final velocity), \( U = 45 \, \text{m/s} \) (initial relative velocity), and \( A = -2 \, \text{m/s}^2 \) (retardation).

Substituting the values into the equation:

\[ 0 = (45)^2 + 2(-2)S \]

\[ 0 = 2025 - 4S \]

\[ 4S = 2025 \]

\[ S = \frac{2025}{4} = 506.25 \, \text{m} \]

### Step 5: Calculate Remaining Distance
Initially, the trains are \( 1000 \, \text{m} \) apart. After traveling \( 506.25 \, \text{m} \), the distance remaining between them is:

\[ \text{Remaining Distance} = 1000 \, \text{m} - 506.25 \, \text{m} = 493.75 \, \text{m} \]

### Step 6: Conclusion
Since there is still \( 493.75 \, \text{m} \) of distance remaining after both trains have applied brakes, they will not collide.

### Final Answer
**No, the trains will not collide.**

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