Solving for v, we getv=\sqrt{\frac{2}{m}\int_{x_1} ^{x_2} F dx}

In summary, the conversation is discussing the calculation of the kinetic energy of a 2 kg book that is initially at rest on a flat frictionless surface. A force of (2.5-x2) is applied to the book, causing it to move through x=2 m. The conversation mentions using the equations W= Int (force) and K= (1/2)mv2 to find the kinetic energy, and the importance of considering the starting and ending positions of the book. The final summary is that the integral of the force over the given distance is equal to the kinetic energy of the book.
  • #1
ssm11s
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1. A 2 kg book is at rest on a flat frictionless surface at initial position xi=0. A force F= (2.5-x2) is applied on the book. What is the Kinetic Energy of the block as it passes through x=2 m ?



2. W= Int ( force ) ; K= (1/2)mv2 ; W = K1 - K2



3. So I started out with finding the W by taking the integral of F from x=0 to x=2. Then i put the value of the work in the work-kinetic energy equation. And I am stuck here cause i don't know where to use the mass of the book.
 
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  • #2
Well now, the value of your integral is the kinetic energy.

[tex]Change \ in \ kinetic \ energy = \int_{x_1} ^{x_2} F dx[/tex]


Since it started at rest, the initial ke is 0 and the final one is 1/2mv2

so

[tex]\frac{1}{2}mv^2-0=\int_{x_1} ^{x_2} F dx[/tex]
 
  • #3


As a scientist, your response should be clear and concise while also addressing the specific concerns and questions mentioned in the content. Here is a possible response:

The equation provided, v = √(2/m ∫Fdx), is used to solve for the velocity (v) of an object given its mass (m) and the force (F) acting on it over a certain distance (x1 to x2). In this scenario, the force acting on the 2 kg book is given by F = (2.5-x^2), and we are interested in finding the kinetic energy of the book as it moves from x=0 to x=2 m.

To solve for the kinetic energy, we first need to find the velocity of the book at x=2 m. This can be done by substituting the given values into the equation: v = √(2/2 ∫(2.5-x^2)dx). Solving the integral gives us v = √(5-2x^2). Plugging in x=2, we get v=2 m/s.

Next, we can use the kinetic energy equation, K = (1/2)mv^2, to calculate the kinetic energy of the book at x=2 m. We know the mass of the book is 2 kg, and we just found the velocity to be 2 m/s. Thus, K = (1/2)(2 kg)(2 m/s)^2 = 2 J.

In conclusion, the kinetic energy of the 2 kg book at x=2 m is 2 J. The mass of the book is incorporated into the equation for velocity, which is then used to calculate the kinetic energy. I hope this explanation helps clarify the process for solving this problem.
 

1. What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion. It is calculated by the formula KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.

2. How is kinetic energy different from potential energy?

Kinetic energy is the energy of motion, while potential energy is the energy an object possesses due to its position or state. An object can have both kinetic and potential energy at the same time.

3. What is the relationship between kinetic energy and work?

Work is defined as the force applied to an object multiplied by the distance it travels. The work done on an object is equal to its change in kinetic energy. This means that when work is done on an object, its kinetic energy increases or decreases depending on the direction of the force.

4. Can kinetic energy be converted into other forms of energy?

Yes, kinetic energy can be converted into other forms of energy, such as potential energy, thermal energy, and sound energy. This is demonstrated by the conservation of energy principle, which states that energy cannot be created or destroyed, only transformed from one form to another.

5. How does the mass and velocity of an object affect its kinetic energy?

The mass of an object affects its kinetic energy directly, meaning that the more massive an object is, the more kinetic energy it possesses. The velocity of an object affects its kinetic energy exponentially, meaning that doubling the velocity will quadruple the kinetic energy. This shows that velocity has a greater impact on kinetic energy than mass.

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