Question

A car goes from one end to the other end of a semicircular path of diameter ' \(\mathrm{d}\) '. Find the ratio between path length and displacement. $\begin{array}{lll}\text { (A) }(3 \pi / 2) & \text { (B) } \pi \text { (C) } 2 & \text { (D) } \pi / 2\end{array}$

Short answer

The ratio between the path length and displacement is \(\frac{\pi}{2}\).

Step-by-step solution

Calculate the path length

The car is traveling along a semicircular path with a diameter d. To find the path length or distance traveled by the car, we need to find half the circumference of the circle with diameter d. The formula for the circumference of a circle is given by: \[ C = \pi d\] Since the car is traveling along a semicircle, the path length (L) will be half the circumference: \[ L = \frac{1}{2} \pi d\]

Calculate the displacement

Displacement is the straight-line distance between the starting and ending points of the path. In this case, the starting and ending points lie on the diameter which is the straight line passing through the center of the semicircle. Therefore, the displacement (D) is equal to the diameter, d: \[ D = d\]

Find the ratio between path length and displacement

Now, we need to find the ratio between the path length (L) and the displacement (D). This can be calculated by dividing the path length by the displacement: \[ \text{Ratio} = \frac{L}{D}\] Substitute L and D with the values obtained from Step 1 and Step 2: \[ \text{Ratio} = \frac{\frac{1}{2} \pi d}{d}\]

Simplify the ratio

Now, we need to simplify the ratio obtained in Step 3. We can do so by canceling out the common factor 'd' from both numerator and denominator: \[ \text{Ratio} = \frac{\frac{1}{2} \pi d}{d} \times \frac{1}{d} = \frac{\frac{1}{2} \pi}{1} = \frac{\pi}{2}\] Thus, the ratio between the path length and displacement is \(\frac{\pi}{2}\). The correct answer is (D).