A 3.00 kg model rocket is launched straight up. It reaches a maximum height of `1.00xx10^(2)` m above where its engine cuts out, even though air resistance performs `-8.00xx10^(2)J` of work on the rocket. What would have been this height if there were no air resistance ?

Options

A. 111m
B. 135m
C. 159m
D. 127m

Answer

To solve the problem of how high the model rocket would have gone without air resistance, we can follow these steps:

### Step 1: Identify the known values
- Mass of the rocket (m) = 3.00 kg
- Maximum height reached with air resistance (h_a) = 1.00 x 10^2 m = 100 m
- Work done by air resistance (W_air) = -8.00 x 10^2 J = -800 J
- Acceleration due to gravity (g) = 9.8 m/s²

### Step 2: Calculate the work done against gravity when the rocket reaches 100 m
The work done against gravity (W_gravity) when the rocket reaches a height of 100 m can be calculated using the formula:
\[ W_{\text{gravity}} = -mgh_a \]
Substituting the values:
\[ W_{\text{gravity}} = - (3.00 \, \text{kg}) \times (9.8 \, \text{m/s}^2) \times (100 \, \text{m}) \]
\[ W_{\text{gravity}} = -2940 \, \text{J} \]

### Step 3: Apply the work-energy principle
According to the work-energy principle, the total work done on the rocket is equal to the change in kinetic energy. Since the rocket starts from rest and comes to rest at the maximum height, the change in kinetic energy (ΔK) is zero.

Thus, we can express the total work done (W_total) as:
\[ W_{\text{total}} = W_{\text{engine}} + W_{\text{gravity}} + W_{\text{air}} = 0 \]
Rearranging gives:
\[ W_{\text{engine}} = -W_{\text{gravity}} - W_{\text{air}} \]
Substituting the values:
\[ W_{\text{engine}} = 2940 \, \text{J} - 800 \, \text{J} \]
\[ W_{\text{engine}} = 2140 \, \text{J} \]

### Step 4: Calculate the new height without air resistance
If there were no air resistance, all the work done by the engine would go into lifting the rocket against gravity. Thus:
\[ W_{\text{engine}} = mgh \]
Where h is the new height without air resistance. Rearranging gives:
\[ h = \frac{W_{\text{engine}}}{mg} \]
Substituting the values:
\[ h = \frac{2140 \, \text{J}}{(3.00 \, \text{kg}) \times (9.8 \, \text{m/s}^2)} \]
\[ h = \frac{2140}{29.4} \]
Calculating gives:
\[ h \approx 72.8 \, \text{m} \]

### Step 5: Calculate the total height without air resistance
Now, to find the total height without air resistance, we add the height reached with the engine's work against gravity:
\[ H = h_a + h \]
Substituting the values:
\[ H = 100 \, \text{m} + 72.8 \, \text{m} \]
\[ H \approx 172.8 \, \text{m} \]

### Final Answer
Thus, the maximum height the rocket would have reached without air resistance is approximately **172.8 m**.

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