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The work done against gravity (= increase in potential energy) is the same in both cases. Extra work might be done to create kinetic energy.

If the walker and runner both start and end at rest, then overall no kinetic energy is created in either case. All of the kinetic energy which is created is used up to increase the potential energy.

If the walker is still walking and the runner is still running at the top of the stairs then overall some kinetic energy has been created. The runner has created more kinetic energy than the walker because he/she is moving faster at the finish.

$$\text{total work done by person = increase in potential energy + increase in kinetic energy}$$

The person who runs up the stairs finishes with a greater increase in kinetic energy, so the total work done by the runner is greater.

Note that it is the overall increase in kinetic energy between start and finish which matters. It makes no difference how the kinetic energy has varied in between. In physics terms, person A who walks very slowly for most of the way then runs the last few steps does more work than person B who runs most of the way then walks the last few steps.

Person B will probably feel more tired than person A, but this is a separate issue about the difference between external and internal work. Person B does less external work (because he/she has created less kinetic energy) and more internal work (because he/she has been more active and as a result gotten hotter). In physics (mechanics) we are usually only concerned about external work.

The work done against gravity (= increase in potential energy) is the same in both cases. Extra work is done if kinetic energy is increased.

If the walker and runner both start and end at rest (or at the same speed), then there is no overall increase in kinetic energy in either case. Any kinetic energy which is created by accelerating in between is used up again during deceleration to increase the potential energy.

If the walker and runner are moving faster at the top of the stairs than at the bottom then there is some increase in kinetic energy in both cases. Presumably the runner's increase in kinetic energy is greater.

$$\text{total work done by person = increase in potential energy + increase in kinetic energy}$$

The runner finishes with a greater increase in kinetic energy, so the total work done by the runner is greater.

Note that it is the overall increase in kinetic energy between start and finish which matters. It makes no difference how the kinetic energy has varied in between. In mechanical terms, person A who walks slowly for most of the way then runs the last few steps does more work than person B who runs most of the way then walks slowly the last few steps.

Person B uses more energy than person A, but this is a separate issue about the difference between external and internal work. Person B does less external work (he/she has created less kinetic energy) but more internal work (he/she has been more active and as a result gotten hotter). In mechanics we are usually only concerned about external work.

The work done against gravity (= increase in potential energy) is the same in both cases. Extra work might be done to create kinetic energy.

If the walker and runner both start and end at rest, then overall no kinetic energy is created in either case. All of the kinetic energy which is created is used up to increase the potential energy.

If the walker is still walking and the runner is still running at the top of the stairs then overall some kinetic energy has been created. The runner has created more kinetic energy than the walker because he/she is moving faster at the finish.

$$\text{total work done by person = increase in potential energy + increase in kinetic energy}$$

The person who runs up the stairs finishes with a greater increase in kinetic energy, so the total work done by the person is greater.

Note that it is the overall increase in kinetic energy between start and finish which matters. It makes no difference how the kinetic energy has varied in between. In physics terms, person A who walks very slowly for most of the way then runs the last few steps does more work than person B who runs most of the way then walks the last few steps.

Person B will probably feel more tired than person A, but this is a separate issue about the difference between external and internal work. Person B does less external work (because he/she has created less kinetic energy) and more internal work (because he/she has been more active and as a result gotten hotter). In physics (mechanics) we are usually only concerned about external work.

The work done against gravity (= increase in potential energy) is the same in both cases. Extra work might be done to create kinetic energy.

If the walker and runner both start and end at rest, then overall no kinetic energy is created in either case. All of the kinetic energy which is created is used up to increase the potential energy.

If the walker is still walking and the runner is still running at the top of the stairs then overall some kinetic energy has been created. The runner has created more kinetic energy than the walker because he/she is moving faster at the finish.

$$\text{total work done by person = increase in potential energy + increase in kinetic energy}$$

The person who runs up the stairs finishes with a greater increase in kinetic energy, so the total work done by the runner is greater.

Note that it is the overall increase in kinetic energy between start and finish which matters. It makes no difference how the kinetic energy has varied in between. In physics terms, person A who walks very slowly for most of the way then runs the last few steps does more work than person B who runs most of the way then walks the last few steps.

Person B will probably feel more tired than person A, but this is a separate issue about the difference between external and internal work. Person B does less external work (because he/she has created less kinetic energy) and more internal work (because he/she has been more active and as a result gotten hotter). In physics (mechanics) we are usually only concerned about external work.

The work done against gravity (= increase in potential energy) is the same in both cases. Extra work might be done to create kinetic energy.

If the walker and runner both start and end at rest, then overall no kinetic energy is created in either case. All of the kinetic energy which is created is used up to increase the potential energy.

If the walker is still walking and the runner is still running at the top of the stairs then overall some kinetic energy has been created. The runner has created more kinetic energy than the walker because he/she is moving faster at the finish.

$$\text{total work done by person = increase in potential energy + increase in kinetic energy}$$

The person who runs up the stairs finishes with a greater increase in kinetic energy, so the total work done by the person is greater.

Note that it is the overall increase in kinetic energy between start and finish which matters. It makes no difference how the kinetic energy has varied in between. In physics terms, person A who walks very slowly for most of the way then runs the last few steps does more work than person B who runs most of the way then walks the last few steps. Person B will probably be more tired than person A, but this is a separate issue about the difference between external physical work and internal biological work. See Why does holding something up cost energy while no work is being done?

The work done against gravity (= increase in potential energy) is the same in both cases. Extra work might be done to create kinetic energy.

If the walker and runner both start and end at rest, then overall no kinetic energy is created in either case. All of the kinetic energy which is created is used up to increase the potential energy.

If the walker is still walking and the runner is still running at the top of the stairs then overall some kinetic energy has been created. The runner has created more kinetic energy than the walker because he/she is moving faster at the finish.

$$\text{total work done by person = increase in potential energy + increase in kinetic energy}$$

The person who runs up the stairs finishes with a greater increase in kinetic energy, so the total work done by the person is greater.

Note that it is the overall increase in kinetic energy between start and finish which matters. It makes no difference how the kinetic energy has varied in between. In physics terms, person A who walks very slowly for most of the way then runs the last few steps does more work than person B who runs most of the way then walks the last few steps. 

Person B will probably feel more tired than person A, but this is a separate issue about the difference between external and internal work. Person B does less external work (because he/she has created less kinetic energy) and more internal work (because he/she has been more active and as a result gotten hotter). In physics (mechanics) we are usually only concerned about external work.