A small 8.00 kgkg rocket burns fuel that exerts a time-varying upward force on the rocket (assume constant mass) as the rocket moves upward from the launch pad. This force obeys the equation F=A+Bt2𝐹=𝐴+𝐵𝑡2. Measurements show that at t=0𝑡=0, the force is 100.0 NN, and at the end of the first 2.00 ss, it is 168.0 NN.

Find the acceleration of this rocket at the instant after the fuel ignites.

Express your answer with the appropriate units.

We are tasked with finding the rocket's acceleration immediately after the fuel ignites (at \( t = 0 \)).

### Step 1: Understand the force equation

The force exerted on the rocket is given by the equation:
\[
F(t) = A + Bt^2
\]
where:
- \( A \) and \( B \) are constants,
- \( t \) is the time in seconds.

We are also told that:
- At \( t = 0 \), the force is 100.0 N:
\[
F(0) = 100.0 \, \text{N}
\]
- At \( t = 2.00 \, \text{s} \), the force is 168.0 N:
\[
F(2) = 168.0 \, \text{N}
\]

### Step 2: Determine constants \( A \) and \( B \)

From the first condition, at \( t = 0 \):
\[
F(0) = A + B(0)^2 = A = 100.0 \, \text{N}
\]
Thus, \( A = 100.0 \, \text{N} \).

Now, using the second condition, at \( t = 2.00 \, \text{s} \):
\[
F(2) = 100.0 + B(2)^2 = 168.0
\]
\[
100.0 + 4B = 168.0
\]
\[
4B = 68.0
\]
\[
B = 17.0 \, \text{N/s}^2
\]

Thus, the force equation becomes:
\[
F(t) = 100.0 + 17.0t^2
\]

### Step 3: Find the acceleration at \( t = 0 \)

The net force on the rocket at any time is given by:
\[
F_{\text{net}} = F(t) - mg
\]
where \( m \) is the mass of the rocket (8.00 kg), and \( g \) is the acceleration due to gravity (9.81 m/s²).

At \( t = 0 \):
\[
F_{\text{net}} = 100.0 \, \text{N} - (8.00 \, \text{kg})(9.81 \, \text{m/s}^2)
\]
\[
F_{\text{net}} = 100.0 - 78.48 = 21.52 \, \text{N}
\]

Using Newton's second law \( F = ma \), we can find the acceleration:
\[
a = \frac{F_{\text{net}}}{m} = \frac{21.52 \, \text{N}}{8.00 \, \text{kg}} = 2.69 \, \text{m/s}^2
\]

### Final Answer:
The acceleration of the rocket at the instant after the fuel ignites is \( \boxed{2.69 \, \text{m/s}^2} \).

Would you like further clarification or details on this? Here are five related questions for additional practice:

1. What is the force exerted on the rocket at \( t = 1 \, \text{s} \)?
2. How would the force change if the mass of the rocket were 10.00 kg instead of 8.00 kg?
3. What is the rocket's acceleration at \( t = 2 \, \text{s} \)?
4. If gravity on a different planet were 12 m/s², how would the acceleration change at \( t = 0 \)?
5. How does the time-varying force affect the rocket's motion compared to a constant force?

**Tip**: Always ensure units are consistent throughout a problem to avoid errors in calculations.

A small 8.00 kg rocket burns fuel that exerts a time-varying upward force on the rocket (assume constant mass) as the rocket moves upward from the launch pad. This force obeys the equation F = A + Bt^2. Measurements show that at t = 0, the force is 100.0 N, and at the end of the first 2.00 s, it is 168.0 N. Find the acceleration of this rocket at the instant after the fuel ignites. Express your answer with the appropriate units.

Learn how to calculate the acceleration of a rocket with a time-varying force given by F = A + Bt^2. This example involves solving for constants, applying Newton's Second Law, and understanding the effects of gravity. Suitable for students in Grades 10-12.

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