Negative work on a moving body

[alba]

Suppose the box (stone, bullet..., $$m =1$$) is not moving upward because of a lift, but has been shot and has $$v = 20m/s$$ and $$KE = 200 J$$

Gravity is doing negative work and subtracting energy:
$$F = -ma = 10 \cdot 1 = -10 N$$
We cannot use the palallelogram here as KE is not a force, how do you deal with this situation, how do you describe it mathematically?

It is trivial that the box will reach the height KE/ F = 20 m and that there KE will be 0 and PE = 200, and at any height h $KE_h = 200 -PE_h$, but how do you describe the relation, process, the action of negative work done on the body?
The same problem, I suppose, arise if we try to describe the flight of an arrow, on which drag does negative work.

 

[Doc Al]

You can describe the action of the forces (gravity in your example) either dynamically (using Newton's 2nd law) or energetically (using work done). You'll get the same answer either way.

Dynamically, since the only force acting is gravity you know the resulting acceleration is -g. Using work, F*d is negative since the force (down) and displacement (up) are in opposite directions.

 

[alba]

You can describe the action of the forces (gravity in your example) either dynamically (using Newton's 2nd law) or energetically (using work done). You'll get the same answer either way.

Dynamically, since the only force acting is gravity you know the resulting acceleration is -g. Using work, F*d is negative since the force (down) and displacement (up) are in opposite directions.

Thank you, sir, but it is rather hard for me to understand that, in general terms. Can you refer to the concrete example above? In the case of forces we use the parallelogram and subtract ( if upward F were 15 N we would know = + 5N) in this case how do you descrbe mathematically this opposition? Can we only use the makeshift of mechanical energy?

 

[Doc Al]

In the case of forces we use the parallelogram and subtract ( if upward F were 15 N we would know = + 5N) in this case how do you descrbe mathematically this opposition?

By 'parallelogram', I assume you mean vector addition to find the net force. In your example you only have one force, gravity. F = -mg

 

[PJC01]

In this scenario, we can describe the negative work being done on the moving body by the force of gravity using the formula W = Fd, where W is the work done, F is the force, and d is the distance traveled. In this case, the force of gravity is pulling the box downwards with a force of 10 N, while the box is moving with a velocity of 20 m/s. This results in a negative work of -200 J being done on the box.

To describe this mathematically, we can use the equation W = ΔKE, where W is the work done and ΔKE is the change in kinetic energy. In this case, since the kinetic energy is decreasing from 200 J to 0 J, the work done by gravity is equal to -200 J.

In terms of the relation and process, we can say that the negative work done by gravity is causing a decrease in the kinetic energy of the box, which in turn is being converted into potential energy as the box gains height. This process continues until all of the kinetic energy is converted into potential energy and the box reaches its maximum height, at which point the kinetic energy is 0 and the potential energy is 200 J.

Similarly, for the flight of an arrow, the drag force would do negative work on the arrow, causing a decrease in its kinetic energy and a corresponding increase in potential energy. This process would continue until the arrow reaches its maximum height and all of its kinetic energy is converted into potential energy.

In summary, negative work done on a moving body can be described mathematically using the formula W = Fd or W = ΔKE, and it results in a decrease in the kinetic energy of the body and a corresponding increase in potential energy. The process can be described as a conversion of energy from kinetic to potential as the body moves against a force that is doing negative work.