A bus starts from rest with a constant acceleration of `5 m//s^(2)` at the same time a car travelling with a constant velocity `50 m//s` over takes and passes the bus. How fast is the bus travelling when they are side by side?

Options

A. `10 m//s`
B. `50 m//s`
C. `100 m//s`
D. `150 m//s`

Explanation

To solve the problem, we need to find out how fast the bus is traveling when it is side by side with the car. Here is a step-by-step solution:

### Step 1: Identify the given data
- The bus starts from rest, so its initial velocity (u) = 0 m/s.
- The acceleration of the bus (a) = 5 m/s².
- The car travels with a constant velocity (v_car) = 50 m/s.

### Step 2: Write the equations for distance traveled
- The distance traveled by the car (d_car) in time t can be expressed as:
\[ d_{car} = v_{car} \cdot t = 50t \]
- The distance traveled by the bus (d_bus) under constant acceleration can be expressed as:
\[ d_{bus} = ut + \frac{1}{2} a t^2 = 0 + \frac{1}{2} \cdot 5 \cdot t^2 = \frac{5}{2} t^2 \]

### Step 3: Set the distances equal
Since both the bus and the car are side by side at the same time, their distances will be equal:
\[ d_{car} = d_{bus} \]
\[ 50t = \frac{5}{2} t^2 \]

### Step 4: Rearrange the equation
To solve for t, we rearrange the equation:
\[ \frac{5}{2} t^2 - 50t = 0 \]
Factoring out t gives:
\[ t \left(\frac{5}{2} t - 50\right) = 0 \]
This gives us two solutions: \(t = 0\) (the starting point) or \(\frac{5}{2} t - 50 = 0\).

### Step 5: Solve for t
Solving \(\frac{5}{2} t - 50 = 0\):
\[ \frac{5}{2} t = 50 \]
\[ t = 50 \cdot \frac{2}{5} = 20 \text{ seconds} \]

### Step 6: Calculate the speed of the bus at t = 20 seconds
Now we can find the speed of the bus when t = 20 seconds using the formula:
\[ v = u + at \]
Substituting the values:
\[ v = 0 + 5 \cdot 20 = 100 \text{ m/s} \]

### Conclusion
The speed of the bus when it is side by side with the car is **100 m/s**.

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