Two motor cars start from A simultaneously & reach B after 2 hour. The first car travelled half the distance at a speed of `v_(1)=30 km hr^(-1)` & the other half at a speed of `v_(2)=60 km hr^(-1)`. The second car covered the entire with a constant acceleration. At what instant of time, were the speeds of both the vehicles same? Will one of them overtake the other enroute?

Explanation

To solve the problem, we need to analyze the motion of both cars step by step.

### Step 1: Determine the total distance (d) between points A and B.
The first car travels half the distance at a speed of \( v_1 = 30 \, \text{km/hr} \) and the other half at \( v_2 = 60 \, \text{km/hr} \). Since both cars take 2 hours to reach point B, we can set up the equation for the first car.

Let the total distance \( d \) be divided into two halves, \( \frac{d}{2} \).

The time taken to cover the first half:
\[ t_1 = \frac{\frac{d}{2}}{v_1} = \frac{\frac{d}{2}}{30} = \frac{d}{60} \, \text{hours} \]

The time taken to cover the second half:
\[ t_2 = \frac{\frac{d}{2}}{v_2} = \frac{\frac{d}{2}}{60} = \frac{d}{120} \, \text{hours} \]

The total time taken by the first car is:
\[ t_1 + t_2 = \frac{d}{60} + \frac{d}{120} = \frac{2d + d}{120} = \frac{3d}{120} = \frac{d}{40} \]

Since both cars take 2 hours to reach B:
\[ \frac{d}{40} = 2 \implies d = 80 \, \text{km} \]

### Step 2: Determine the acceleration of the second car.
The second car travels the entire distance \( d \) with constant acceleration. Using the second equation of motion:
\[ d = ut + \frac{1}{2} a t^2 \]
where \( u = 0 \) (initial velocity), \( d = 80 \, \text{km} \), and \( t = 2 \, \text{hours} = 2 \times 3600 \, \text{seconds} = 7200 \, \text{seconds} \).

Substituting the values:
\[ 80 = 0 + \frac{1}{2} a (7200)^2 \]
\[ 80 = \frac{1}{2} a (51840000) \]
\[ a = \frac{80 \times 2}{51840000} = \frac{160}{51840000} \approx 0.00000308 \, \text{km/s}^2 \]
Converting to km/hr²:
\[ a = 0.00000308 \times 3600^2 \approx 40 \, \text{km/hr}^2 \]

### Step 3: Determine when the speeds of both vehicles are the same.
The speed of the first car after time \( t \) (in hours) is:
- For the first half (0 to \( \frac{d}{2} \)):
\[ v_1(t) = 30 \, \text{km/hr} \quad \text{(for } t \leq \frac{d}{60} \text{)} \]
- For the second half (after \( \frac{d}{60} \)):
\[ v_1(t) = 60 \, \text{km/hr} \quad \text{(for } t > \frac{d}{60} \text{)} \]

The speed of the second car after time \( t \) is:
\[ v_2(t) = a \cdot t = 40t \, \text{km/hr} \]

Setting the speeds equal:
1. For the first half:
\[ 30 = 40t \implies t = \frac{3}{4} \, \text{hours} \]

2. For the second half:
\[ 60 = 40t \implies t = \frac{3}{2} \, \text{hours} \]

### Step 4: Determine if one car overtakes the other.
The first car travels at a constant speed of 30 km/hr for the first half and 60 km/hr for the second half. The second car accelerates from rest.

- At \( t = \frac{3}{4} \) hours, the first car is still traveling at 30 km/hr, while the second car has reached:
\[ v_2\left(\frac{3}{4}\right) = 40 \times \frac{3}{4} = 30 \, \text{km/hr} \]

- At \( t = \frac{3}{2} \) hours, the first car is at 60 km/hr, while the second car has reached:
\[ v_2\left(\frac{3}{2}\right) = 40 \times \frac{3}{2} = 60 \, \text{km/hr} \]

Thus, both cars have the same speed at \( t = \frac{3}{4} \) hours and \( t = \frac{3}{2} \) hours.

### Conclusion:
1. The speeds of both vehicles are the same at \( t = \frac{3}{4} \) hours and \( t = \frac{3}{2} \) hours.
2. Yes, one car will overtake the other en route.