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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+Bt2. 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

Expert verified
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

01

Identify Given Variables and Equations

We need to find constants A and B given the force equation F=A+Bt2. The force measurements are F(0)=100.0 N and F(2)=150.0 N.
02

Solve for Constant A

Using the equation for t=0, substitute values into the equation: F=A+B(0)2=100NThis simplifies to A=100.0 N.
03

Solve for Constant B

Using t=2s, substitute into the equation:F=A+B(2)2=150NSubstitute A=100 N:150=100+4BSolve for B:B=504=12.5N/s2
04

Calculate Net Force and Acceleration at t=0

At t=0, with gravity acting downwards:Net force=Fmg=100(8×9.8)=21.6NAcceleration a is given by:a=Net Forcem=21.68=2.7m/s2
05

Calculate Net Force and Acceleration at t=3

At t=3s:F=100+12.5×(3)2=100+112.5=212.5NNet force:Net force=212.5(8×9.8)=134.9NAcceleration:a=134.98=16.86m/s2
06

Calculate Acceleration in Space at t=3

In space, only the force F affects the rocket:Acceleration=Fm=212.58=26.56m/s2

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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+Bt2. 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 Bt2 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 =Fmg
Here, F is the upward force from the rocket, m is the rocket's mass, and g is the acceleration due to gravity, approximately 9.8m/s2 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 Bt2.
  • 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=extnetforcem), 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.

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Most popular questions from this chapter

An astronaut's pack weighs 17.5 N when she is on the earth but only 3.24 N when she is at the surface of a moon. (a) What is the acceleration due to gravity on this moon? (b) What is the mass of the pack on this moon?

The upward normal force exerted by the floor is 620 N on an elevator passenger who weighs 650 N. What are the reaction forces to these two forces? Is the passenger accelerating? If so, what are the magnitude and direction of the acceleration?

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A 75.0-kg man steps off a platform 3.10 m above the ground. He keeps his legs straight as he falls, but his knees begin to bend at the moment his feet touch the ground; treated as a particle, he moves an additional 0.60 m before coming to rest. (a) What is his speed at the instant his feet touch the ground? (b) If we treat the man as a particle, what is his acceleration (magnitude and direction) as he slows down, if the acceleration is assumed to be constant? (c) Draw his freebody diagram. In terms of the forces on the diagram, what is the net force on him? Use Newton's laws and the results of part (b) to calculate the average force his feet exert on the ground while he slows down. Express this force both in newtons and as a multiple of his weight.

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