Solving Quadratic Drag Force on Upward Motion of Gun

[jesuslovesu]

Homework Statement

A gun is fired straight up. Assume a quadratic drag force.
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(upward motion) $$v^2 = Ae^{-2kx} - \frac{g}{k}$$
(downward motion) $$v^2 = g/k - Be^{2kx}$$

Homework Equations

The Attempt at a Solution

I can easily solve the integrals unfortunately I am having difficulty setting equations up.

Upward:
ma = mg + cv^2

Unfortunately, to get the book's answer I would need to say ma = -mg - cv^2
And I am having a difficult time seeing why both terms are negative.

I draw the free body diagram. As soon as the bullet leaves the barrel, the acceleration is in the downward direction. (so mg is pos) and the drag also acts in the downward direction so I thought ma = mg + cv^2. Shouldn't mg and cv^2 both be positive because they are in the direction of the acceleration?

Does anyone know why I am getting different signs?

 

[Dale]

Upward:
ma = mg + cv^2

Unfortunately, to get the book's answer I would need to say ma = -mg - cv^2
And I am having a difficult time seeing why both terms are negative.

I draw the free body diagram. As soon as the bullet leaves the barrel, the acceleration is in the downward direction. (so mg is pos) and the drag also acts in the downward direction so I thought ma = mg + cv^2. Shouldn't mg and cv^2 both be positive because they are in the direction of the acceleration?

Does anyone know why I am getting different signs?

They just defined the positive direction as upwards where you defined it as downwards. There is nothing wrong with either choice, as long as you are consistent while working the problem.

 

[baba89]

I would like to provide a response to the content by first acknowledging the difficulty in setting up the equations and understanding the signs in the equations. It is important to carefully consider the direction of the forces and the direction of the acceleration in order to correctly set up the equations.

In this case, the upward motion of the gun is being affected by both gravity and drag force. Gravity is acting in the downward direction, while the drag force is acting in the opposite direction of the velocity of the bullet. This means that in the upward motion, the drag force will have a negative sign in the equation. This can be seen in the downward motion equation as well, where the drag force is acting in the same direction as the velocity and therefore has a positive sign.

It is important to note that the signs in the equations may vary depending on the chosen coordinate system. However, in this case, it is important to use a consistent coordinate system throughout the problem in order to avoid confusion.

In summary, the negative sign in the drag force term in the upward motion equation is correct and reflects the direction of the drag force acting on the bullet. It is important to carefully consider the direction of the forces and the acceleration in order to correctly set up the equations and solve the problem.