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. none

Answer

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’s the step-by-step solution:

### Step 1: Identify the given data
- The bus starts from rest, so its initial velocity \( u_b = 0 \, m/s \).
- The acceleration of the bus \( a_b = 5 \, m/s^2 \).
- The car travels with a constant velocity \( v_c = 50 \, m/s \).

### Step 2: Set up the equations for distance traveled
Since both the bus and the car will cover the same distance when they are side by side, we can set up the equations for the distances traveled by each.

- The distance traveled by the bus after time \( t \) is given by the equation:
\[ S_b = u_b t + \frac{1}{2} a_b t^2 = 0 + \frac{1}{2} \cdot 5 \cdot t^2 = \frac{5}{2} t^2 \]

- The distance traveled by the car after time \( t \) is:
\[ S_c = v_c t = 50t \]

### Step 3: Set the distances equal
Since both vehicles cover the same distance when they are side by side, we can equate the two distances:
\[ \frac{5}{2} t^2 = 50t \]

### Step 4: Solve for time \( t \)
To solve for \( t \), we can rearrange the equation:
\[ \frac{5}{2} t^2 - 50t = 0 \]
Factoring out \( t \):
\[ t \left( \frac{5}{2} t - 50 \right) = 0 \]
This gives us two solutions:
1. \( t = 0 \) (initial time)
2. \( \frac{5}{2} t - 50 = 0 \) leads to:
\[ \frac{5}{2} t = 50 \implies t = \frac{50 \cdot 2}{5} = 20 \, seconds \]

### Step 5: Find the velocity of the bus at time \( t = 20 \, seconds \)
Now, we can find the velocity of the bus when they are side by side using the formula:
\[ v_b = u_b + a_b t = 0 + 5 \cdot 20 = 100 \, m/s \]

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

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