# Integrating Force: Derive Distance L for AP Question

• aber
In summary, the conversation is discussing the process of deriving an expression for the distance that a dart penetrates into a block of wood when it is shot into it with a certain initial velocity and a force that is proportional to that velocity. The participants mention using integration and breaking the problem up into small intervals of time to approximate the change in velocity and force. They also mention using Newton's 2nd law and the kinematic equation for uniformly accelerated motion.
aber
This last part of an AP questions is giving me some trouble, mostly because i involves integrating and i never took Calculus.

Part D: The dart is now shot into a block of wood that is in a fixed place. The block exerts a Force F on the dart that is proportional to the dart's Velocity V and in the opposite direction, that is F=-bv, where b is a constant. Derive and expression for the distance L that the dart penetrates into the block, in terms of m (mass), v (initial), and b.

Since the Force -bv is not constant, i can't figure out how to use the kinetic energy 1/2mv^2 to solve for distance, i would think you would need X=Xknot+vknotT+1/2aT^2.

aber said:
Since the Force -bv is not constant, i can't figure out how to use the kinetic energy 1/2mv^2 to solve for distance, i would think you would need X=Xknot+vknotT+1/2aT^2.
As you seem to realize, you need to be able to integrate to solve this problem. That kinematic equation is only good for uniformly accelerated motion, which is not the case here.

If you want to try your hand at integrating, here's a hint: $\int F dt = \Delta (mv)$.

aber,

Here's another way to think about it. You have the equation F = -vb. What does Newton's 2nd law say about F? Can you rewrite what it says as a derivative of v intstead of x?

jdavel: i don't know how to derive...
Doc Al: Would it be change in F dt= change in MV, F= MV-M0/t=MV/t? t=2L/3V, F= 3MV^2/2L

You have learned that acceleration is the rate of change of velocity. Since you know the initial velocity, you know the initial force. That force is going to reduce the velocity. If you break the problem up into small intervals of time, you can get a good approximation of the change in velocity in one interval of time by assuming the intial force is constant for the first interval and calculating the change in velocity resulting from that force applied in that short time. At the end of that time, the dart will have a lower velocity, so you can calculate the reduced force and assume it is constant for the next little interval of time, leading to a lower velocity and a lower force for the next interval, etc., etc. If you know how to use a spreadsheet or program a calculator or computer, you can make the times intervals very small and make the approximation as good as you like.

## 1. What is the concept of "integrating force" in relation to the AP Physics exam?

"Integrating force" refers to the process of finding the distance traveled by an object by integrating the force acting on it over a certain period of time. This concept is commonly tested on the AP Physics exam in the context of kinematics and Newton's laws of motion.

## 2. How do you derive the distance (L) for an AP Physics question involving integrating force?

To derive the distance (L), you will need to use the equation L = ∫F(t)dt, where F(t) represents the force acting on the object and dt represents a small change in time. This integral represents the area under the force vs. time graph, which can be found by using numerical integration techniques or by graphing the function and finding the area geometrically.

## 3. What are some common mistakes to avoid when solving an AP Physics problem involving integrating force?

Some common mistakes to avoid include not properly setting up the integral, incorrectly using units, and not accounting for the direction of the force. It is important to carefully read the problem and identify the given information, as well as clearly defining the variables and their units before attempting to solve.

## 4. Can you provide an example of an AP Physics problem that involves integrating force?

Sure! An example problem could be: "A car traveling at a constant velocity of 10 m/s experiences a constant force of 5 N in the same direction for 3 seconds. What is the total distance traveled by the car during this time?" To solve this, you would use the equation L = ∫F(t)dt, where F(t) = 5 N and dt = 3 s. Integrating from t = 0 to t = 3, you would get L = (5 N)(3 s) = 15 m.

## 5. How can I prepare for the AP Physics exam to ensure I understand the concept of integrating force?

To prepare for the AP Physics exam, it is important to review the key equations and concepts related to motion, forces, and integration. Practice solving problems involving integrating force and make sure you understand the steps involved in each solution. It can also be helpful to seek out additional resources, such as online tutorials or review books, to reinforce your understanding of the concept.

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