A model train of mass 0.5 kg is placed at one end of a straight 3 -m electric track. Assume that

step 1 So here I have written down the work energy theorem from exercise 39, which says the amount of w

A model train of mass 0.5 kg is placed at one end of a...

A model train of mass 0.5 $\mathrm{kg}$ is placed at one end of a straight 3 -m electric track. Assume that a force $F(x)=\left(3 x-x^{2}\right)$ N acts on the train at distance $x$ along the track. Use the Work-Energy Theorem (Exercise 39 ) to determine the velocity of the train when it reaches the end of the track.

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Key Concepts

Integration of a Variable Force

When a force varies with position, the work done cannot be calculated by simply multiplying force by displacement. Instead, integration is used to sum the infinitesimal amounts of work done over small displacements. This process involves integrating the force function over the distance traveled, which is crucial in obtaining the total work done on the object.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion, quantified as one half the product of its mass and the square of its velocity. In the context of the Work-Energy Theorem, the change in kinetic energy from one state to another directly relates to the work done, allowing for the determination of velocity after a given amount of work has been applied.

Work-Energy Theorem

The Work-Energy Theorem is a fundamental principle in mechanics that states the net work performed on an object is equal to its change in kinetic energy. This theorem provides a direct connection between the forces acting on an object and the resulting change in its speed, making it central to problems involving energy transfer and motion.

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