Let the original speed be \( x \) km/h. The time taken is \( \frac{360}{x} \) hours.
The new speed is \( x + 5 \) km/h, and the new time taken is \( \frac{360}{x + 5} \) hours.
According to the problem statement, the difference in time is 48 minutes, which is \( \frac{48}{60} = 0.8 \) hours.
Therefore, we have the equation:
\[
\frac{360}{x} - \frac{360}{x+5} = 0.8
\]
Multiply both sides by \( x(x+5) \) to clear the denominators:
\[
360(x+5) - 360x = 0.8x(x+5)
\]
Simplify the equation:
\[
360x + 1800 - 360x = 0.8x^2 + 4x
\Rightarrow 1800 = 0.8x^2 + 4x
\]
To eliminate the decimal, multiply the entire equation by 10:
\[
18000 = 8x^2 + 40x
\Rightarrow 8x^2 + 40x - 18000 = 0
\]
Divide by 8 to simplify:
\[
x^2 + 5x - 2250 = 0
\]
Solve this quadratic equation using the quadratic formula:
\[
x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-2250)}}{2(1)} = \frac{-5 \pm \sqrt{25 + 9000}}{2} = \frac{-5 \pm \sqrt{9025}}{2} = \frac{-5 \pm 95}{2}
\]
This gives two possible solutions for x: \( x = \frac{90}{2} = 45 \) or \( x = \frac{-100}{2} = -50 \).
Since speed cannot be negative, we discard the negative value. Thus, \( x = 45 \).
Let's verify this solution.
Original time = \( \frac{360}{45} = 8 \) hours.
New speed = \( 45 + 5 = 50 \) km/h.
New time = \( \frac{360}{50} = 7.2 \) hours.
The difference in time is \( 8 - 7.2 = 0.8 \) hours, which is equivalent to \( 0.8 \times 60 = 48 \) minutes. This matches the given condition.
The original speed is \(\boxed{45 \text{ km/h}}\).