The distance covered by a body in time `(30.0 pm 0.4)`m is `(6.0pm 0.6)`s. Calculate the speed of the body. The percentage error in the speed is

Answer

To solve the problem step by step, we will calculate the speed of the body and then determine the percentage error in that speed.

### Step 1: Identify the given values
- Distance \( s = 30.0 \pm 0.4 \) m
- Time \( t = 6.0 \pm 0.6 \) s

### Step 2: Calculate the speed
The formula for speed \( v \) is given by:
\[ v = \frac{s}{t} \]
Substituting the values:
\[ v = \frac{30.0 \, \text{m}}{6.0 \, \text{s}} = 5.0 \, \text{m/s} \]

### Step 3: Calculate the uncertainties in speed
To find the uncertainty in speed \( \Delta v \), we use the formula for the propagation of uncertainty:
\[ \frac{\Delta v}{v} = \frac{\Delta s}{s} + \frac{\Delta t}{t} \]
Where:
- \( \Delta s = 0.4 \, \text{m} \)
- \( \Delta t = 0.6 \, \text{s} \)

Substituting the values:
\[ \frac{\Delta v}{5.0} = \frac{0.4}{30.0} + \frac{0.6}{6.0} \]

Calculating each term:
1. \( \frac{0.4}{30.0} \approx 0.01333 \)
2. \( \frac{0.6}{6.0} = 0.1 \)

Adding these:
\[ \frac{\Delta v}{5.0} = 0.01333 + 0.1 = 0.11333 \]

Now, multiply by \( v \) to find \( \Delta v \):
\[ \Delta v = 5.0 \times 0.11333 \approx 0.56665 \, \text{m/s} \approx 0.57 \, \text{m/s} \]

### Step 4: Final speed with uncertainty
Thus, the speed of the body is:
\[ v = 5.0 \pm 0.57 \, \text{m/s} \]

### Step 5: Calculate the percentage error in speed
The percentage error is given by:
\[ \text{Percentage error} = \left( \frac{\Delta v}{v} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage error} = \left( \frac{0.57}{5.0} \right) \times 100 \approx 11.4\% \]

### Final Answer
The speed of the body is \( 5.0 \pm 0.57 \, \text{m/s} \) and the percentage error in the speed is approximately \( 11.4\% \).

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