Question

A small 8.00-kg rocket burns fuel that exerts a timevarying 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 150.0 N. (a) Find the constants \(A\) and \(B\), including their SI units. (b) Find the \(net\) force on this rocket and its acceleration (i) the instant after the fuel ignites and (ii) 3.00 s after the fuel ignites. (c) Suppose that you were using this rocket in outer space, far from all gravity. What would its acceleration be 3.00 s after fuel ignition?

Short answer

A = 100 N, B = 12.5 N/s²; acceleration at t = 0 is 2.7 m/s², at t = 3 s is 16.86 m/s²; in space at t = 3 s is 26.56 m/s².

Step-by-step solution

Identify Given Variables and Equations

We need to find constants \( A \) and \( B \) given the force equation \( F = A + Bt^2 \). The force measurements are \( F(0) = 100.0 \) N and \( F(2) = 150.0 \) N.

Solve for Constant A

Using the equation for \( t = 0 \), substitute values into the equation: \[F = A + B(0)^2 = 100 \, \text{N}\]This simplifies to \( A = 100.0 \) N.

Solve for Constant B

Using \( t = 2 \, \text{s} \), substitute into the equation:\[F = A + B(2)^2 = 150 \, \text{N}\]Substitute \( A = 100 \) N:\[150 = 100 + 4B\]Solve for \( B \):\[B = \frac{50}{4} = 12.5 \, \text{N/s}^2\]

Calculate Net Force and Acceleration at t=0

At \( t = 0 \), with gravity acting downwards:\[\text{Net force} = F - mg = 100 - (8 \times 9.8) = 21.6 \, \text{N}\]Acceleration \( a \) is given by:\[a = \frac{\text{Net Force}}{m} = \frac{21.6}{8} = 2.7 \, \text{m/s}^2\]

Calculate Net Force and Acceleration at t=3

At \( t = 3 \, \text{s} \):\[F = 100 + 12.5 \times (3)^2 = 100 + 112.5 = 212.5 \, \text{N}\]Net force:\[\text{Net force} = 212.5 - (8 \times 9.8) = 134.9 \, \text{N}\]Acceleration:\[a = \frac{134.9}{8} = 16.86 \, \text{m/s}^2\]

Calculate Acceleration in Space at t=3

In space, only the force \( F \) affects the rocket:\[\text{Acceleration} = \frac{F}{m} = \frac{212.5}{8} = 26.56 \, \text{m/s}^2\]

Key concepts

Time-Varying Force

In our study of Newton's Second Law, a key aspect is understanding how forces change over time. This is what we call a time-varying force. In our exercise, the force exerted by the rocket fuel changes as a function of time, expressed by the equation \( F = A + Bt^2 \). Here, both \( A \) and \( B \) are constants that determine how the force behaves at different moments.

**Understanding Time-dependence**:

• When \( t = 0 \), the force is simply \( A \).
• As time progresses, the term \( Bt^2 \) grows, indicating the force increases with time.

The force isn't static; instead, it adapts over time, which is crucial for scenarios like rocket launches, where varied force application affects acceleration and trajectory.

Net Force Calculation

To calculate the net force acting on the rocket, we must account for all existing forces. This typically involves both the force applied by the rocket's fuel and gravitational force. The equation to find the net force is:

• Net force \( = F - mg \)

Here, \( F \) is the upward force from the rocket, \( m \) is the rocket's mass, and \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \) on Earth.

**Net Force in Different Scenarios**:

• At \( t = 0 \), the net force is just the initial force minus the gravitational force.
• As time increases, \( F \) increases because of the term \( Bt^2 \).
• Thus, the net force also increases, affecting acceleration.

By understanding the forces at play, we can predict the motion of the rocket at different time intervals.

Rocket Propulsion

Rocket propulsion is a fascinating application of Newton's Second Law, and it centers around the explosion of fuel to exert force. The time-varying nature of this force is crucial, as it influences how fast and how far a rocket can travel.

**Key Points in Rocket Propulsion**:

• Rockets utilize the force generated from burning fuel to push against the mass of the rocket.
• This propulsion is a direct application of Newton's law: the action of expelled gases results in a reaction force propelling the rocket.
• The higher the net force, the greater the acceleration (\( a = \frac{ ext{net force}}{m} \)), meaning faster propulsion into space.
• In space, without gravitational forces, the only force acting is the propulsion force, maximizing acceleration.

These principles not only explain how rockets escape Earth's gravity, but also how they maneuver in the vacuum of space.