# A hummingbird has a mass of about 1.7 g. Suppose a hummingbird does 0.15 J of work against...

## Question:

A hummingbird has a mass of about {eq}1.7 \ g {/eq}. Suppose a hummingbird does {eq}0.15 \ J {/eq} of work against gravity, so that it ascends straight up with a net acceleration of {eq}1.2 \ m / s {/eq} (squared). How far up does it move?

## Work:

Work is a form of energy transfer due to the motion of an object. Work can be defined in terms of the kinetic energy change that an object can experience, or the applied external force of motion on the object times its distance traveled. Just like with energy values, work is also expressed in joules (J).

Given:

• {eq}\displaystyle \rm m = 1.7\ g = 0.0017\ kg {/eq} is the mass of the bird
• {eq}\displaystyle \rm W = 0.15\ J {/eq} is the work done by the bird against gravity
• {eq}\displaystyle \rm a = 1.2\ m/s^2 {/eq} is the net acceleration of the bird

We can determine the distance traveled by the bird by noting that the definition of work is:

{eq}\displaystyle \rm W = Fd {/eq}

The force F here is the force done by the bird against gravity. This force, in particular, can be determined through using Newton's second law. Note that the force acts against gravity, so we actually have two forces:

{eq}\displaystyle \rm ma = F - mg {/eq}

The force is:

{eq}\displaystyle \rm F = ma + mg {/eq}

{eq}\displaystyle \rm F = m(a + g) {/eq}

We re-write our work:

{eq}\displaystyle \rm W = m(a+g)d {/eq}

We isolate our distance d:

{eq}\displaystyle \rm d = \frac{W}{m(a+g)} {/eq}

We substitute:

{eq}\displaystyle \rm d = \frac{0.15\ J}{(0.0017\ kg)(9.8\ m/s^2 + 1.2\ m/s^2)} {/eq}

We will thus get:

{eq}\displaystyle \rm \boxed{\rm d = 8.02\ m} {/eq}