What is the kinetic energy of the
arrow?
KE = ½ mv^2
KE = ½(0.015 kg)(5.0 m/s)^2
KE = 0.19 J
6.5 - Work-kinetic energy theorem
Work-kinetic energy theorem: The net work
done on a particle equals its change in kinetic
energy.
Consider the foot kicking the soccer ball in Concept 1. We want to relate the work done
by the force exerted by the foot on the ball to the ball’s change in kinetic energy. To
focus solely on the work done by the foot, we ignore other forces acting on the ball,
such as friction.
Initially, the ball is stationary. It has zero kinetic energy because it has zero speed. The
foot applies a force to the ball as it moves through a short displacement. This force
accelerates the ball. The ball now has a speed greater than zero, which means it has
kinetic energy. The work-kinetic energy theorem states that the work done by the foot
on the ball equals the change in the ball’s kinetic energy. In this example, the work is
positive (the force is in the direction of the displacement) so the work increases the
kinetic energy of the ball.
As shown in Concept 2, a goalie catches a ball kicked directly at her. The goalie’s
hands apply a force to the ball, slowing it. The force on the ball is opposite the ball’s
displacement, which means the work is negative. The negative work done on the ball
slows and then stops it, reducing its kinetic energy to zero. Again, the work equals the
change in energy; in this case, negative work on the ball decreases its energy.
In the scenarios described here, the ball is the object to which a force is applied. But
you can also think of the soccer ball doing work. The ball applies a force on the goalie,
causing the goalie’s hands to move backward. The ball does positive work on the goalie
because the force it applies is in the direction of the displacement of the goalie’s hands.
When stated precisely, which is always worthwhile, the work-kinetic energy theorem is
defined to apply to a particle: The net work done on a particle equals the change in its
KE. A particle is a small, indivisible point of mass that does not rotate, deform, and so
on. Various properties of the particle can be observed at any point in time, and by
recording those properties at any instant, its state can be defined.
The only form of energy that a single particle can possess is kinetic energy. Stating the work-kinetic energy theorem for a particle means that
the work contributes solely to the change in the particle’s kinetic energy. A soccer ball is not a particle. When you kick a soccer ball, the
surface of the ball deforms, the air particles inside move faster, and so forth.
Having said this, we (and others) apply the work-kinetic energy theorem to objects such as soccer balls. A textbook filled solely with particles
would be a drab textbook indeed. We simplify the situation, modeling the ball as a particle, so that we can apply the work-kinetic energy
theorem. We can always make it more complicated (have the ball rotate or lift off the ground, so rotational KE and gravitational potential
energy become factors), but the work-kinetic energy theorem provides an essential starting point.
For instance, in Example 1, we first calculate the work done on the ball by the foot. We then use the work-kinetic energy theorem to equate the
work to the change in KE of the ball. Using the definition of KE, we can calculate the ball’s speed immediately after being kicked.
It is important that the theorem applies to the net work done on an object. Here, we ignore the force of friction, but if it were being considered,
we would have to first calculate the net force being applied on the ball in order to consider the net work that is done on it.
Work done on a particle
Net work equals change in kinetic
energy
Positive work on object increases its KENegative work on object
Decreases object’s kinetic energy