When an object is lifted or projected upward, work must be done against the resistance from gravity. In some situations, the resistance of inertia from accelerating the object and air resistance must be taken into account.

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If the object is already moving upward at some initial velocity, the work done by gravity is simply the force of gravity times the displacement, provided the velocity is constant and small enough that air resistance is neg liable.

If the object is accelerated during the lifting process, the resistance from inertia must be taken into account. If the object is projected upward at a high velocity, air resistance must be added to the equation.

Questions you may have include:

What is the work done when the object is lifted at a constant velocity? What is the work when the object is accelerated upward? What happens when the object is projected upward at a high velocity?

This lesson will answer those questions. Useful tool: Units Conversion

If an object has an initial upward velocity and some force is continuing to move it upward at that constant velocity, the force required to move the object equals the restive force of gravity:

FT = Fg

Fg = mg

where

FTis the upward forceFg is the resistive force of gravitym is the mass of the objectg is the acceleration due to gravity

The work required to lift the object is:

W = mgy

where

W is the work requiredy is the displacement of the object

## Accelerating upward

If an object is stationary or moving upward at some initial velocity, a sufficient upward force can accelerate the object to higher velocities, The upward force must be greater than the downward force of gravity in order to accelerate the object upward.

Most often the air resistance in relatively low upward acceleration is negligible.

The upward force is:

FT = Fi + F

FT = ma + mg

where

Fi is the resistive force of inertiaa is the acceleration of the object

Th resulting work required to lift and accelerate an object is:

W = m(a + g)y

If the object is accelerated to some height and then is continued to be lifted at a constant velocity, the work becomes:

W = m(a + g)y1 + mgy2

## High velocity acceleration

When an object, such as a rocket, is acceleration upwards at a high velocity, air resistance becomes an added factor in the work required.

The total upward force to propel a rocket upward is:

FT = Fg + Fi + Fa

FT = mg + ma + kv2

where

Fa is the force of the air resistance or dragk is the air resistance constant related to the size and shape of the objectv is the velocity of the object

The work required to propel a rocket to some altitude is:

WT = FTy

WT = (mg + ma + kv2)y

## Summary

When an object is lifted or projected upward, work must be done against the resistance from gravity, inertia and air resistance.

If the object is moving upward at constant velocity, the work done by gravity is simply the force of gravity times the displacement. If the object is accelerated during the lifting process, the resistance from inertia must be taken into account. If the object is projected upward at a high velocity, air resistance must be added to the equation.

Hard work will get you far in life

## Resources and references

Ron Kurtus" Credentials

### Websites

Work by gravity by Sunil Kumar Singh - Connexions

Gravity and Inertia in Running - Locomotion and Biology paper (PDF)

Gravity Resources

### Books

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