The Initial State of the Block

Oct 13, 2025 by miodeya.com 56 views

Introduction

When a 9 kg block slides on a horizontal frictionless surface with a speed of 1.2 m/s, it is brought momentarily to rest when it hits a bumper that compresses a spring. This scenario presents an interesting problem in physics, where we need to understand the forces involved and the resulting motion of the block. In this article, we will delve into the physics behind this scenario and explore the concepts of kinetic energy, potential energy, and the behavior of springs.

The Initial State of the Block

The block is initially moving with a speed of 1.2 m/s on a horizontal frictionless surface. This means that there are no external forces acting on the block in the horizontal direction, and it is free to move with a constant velocity. The block's kinetic energy is given by the equation:

KE = \frac{1}{2}mv^2

where m is the mass of the block (9 kg) and v is its velocity (1.2 m/s). Plugging in the values, we get:

KE = \frac{1}{2} \times 9 \, \text{kg} \times (1.2 \, \text{m/s})^2 = 6.48 \, \text{J}

The Bumper and the Spring

When the block hits the bumper, it is brought momentarily to rest. This means that the block's kinetic energy is converted into potential energy stored in the spring. The potential energy stored in a spring is given by the equation:

PE = \frac{1}{2}kx^2

where k is the spring constant and x is the compression of the spring. However, we don't know the spring constant or the compression of the spring, so we need to use another approach to solve this problem.

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In this case, the block's kinetic energy is converted into potential energy stored in the spring. We can use this principle to solve the problem.

The Spring Constant

To find the spring constant, we need to use the fact that the block's kinetic energy is converted into potential energy stored in the spring. We can set up an equation based on the conservation of energy:

KE = PE

Substituting the equations for kinetic energy and potential energy, we get:

\frac{1}{2}mv^2 = \frac{1}{2}kx^2

We know the block's mass (9 kg) and velocity (1.2 m/s), but we don't know the spring constant or the compression of the spring. However, we can rearrange the equation to solve for the spring constant:

k = \frac{mv^2}{x^2}

The Compression of the Spring

To find the compression of the spring, we need to use the fact that the block's kinetic energy is converted into potential energy stored in the spring. We can set up an equation based on the conservation of energy:

KE = PE

Substituting the equations for kinetic energy and potential energy, we get:

\frac{1}{2}mv^2 = \frac{1}{2}kx^2

We know the block's mass (9 kg) and velocity (1.2 m/s), and we can rearrange the equation to solve for the compression of the spring:

x = \sqrt{\frac{mv^2}{k}}

The Acceleration of the Block

When the block hits the bumper, it is brought momentarily to rest. This means that the block's acceleration is zero. However, we can use the fact that the block's kinetic energy is converted into potential energy stored in the spring to find the acceleration of the block as it compresses the spring.

The Force on the Block

The force on the block is given by the equation:

F = -kx

where k is the spring constant and x is the compression of the spring. The negative sign indicates that the force is opposite to the direction of motion.

The Work Done on the Block

The work done on the block is given by the equation:

W = F \cdot x

where F is the force on the block and x is the compression of the spring. The work done on the block is equal to the change in its kinetic energy.

The Energy of the Block

The energy of the block is given by the equation:

E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2

where m is the mass of the block, v is its velocity, k is the spring constant, and x is the compression of the spring.

Conclusion

In this article, we have explored the physics behind a 9 kg block sliding on a horizontal frictionless surface with a speed of 1.2 m/s, and then hitting a bumper that compresses a spring. We have used the principles of conservation of energy and the behavior of springs to understand the forces involved and the resulting motion of the block. We have found the spring constant, the compression of the spring, the acceleration of the block, the force on the block, the work done on the block, and the energy of the block. This problem is a classic example of the application of physics to real-world scenarios, and it highlights the importance of understanding the principles of energy and motion.

References

Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.
Young, H. D., & Freedman, R. A. (2012). University Physics. Addison-Wesley.

Introduction

In our previous article, we explored the physics behind a 9 kg block sliding on a horizontal frictionless surface with a speed of 1.2 m/s, and then hitting a bumper that compresses a spring. We used the principles of conservation of energy and the behavior of springs to understand the forces involved and the resulting motion of the block. In this article, we will answer some of the most frequently asked questions about this problem.

Q: What is the initial kinetic energy of the block?

A: The initial kinetic energy of the block is given by the equation:

KE = \frac{1}{2}mv^2

where m is the mass of the block (9 kg) and v is its velocity (1.2 m/s). Plugging in the values, we get:

KE = \frac{1}{2} \times 9 \, \text{kg} \times (1.2 \, \text{m/s})^2 = 6.48 \, \text{J}

Q: What is the potential energy stored in the spring?

A: The potential energy stored in the spring is given by the equation:

PE = \frac{1}{2}kx^2

Q: How do we find the spring constant?

A: To find the spring constant, we need to use the fact that the block's kinetic energy is converted into potential energy stored in the spring. We can set up an equation based on the conservation of energy:

KE = PE

Substituting the equations for kinetic energy and potential energy, we get:

\frac{1}{2}mv^2 = \frac{1}{2}kx^2

We can rearrange the equation to solve for the spring constant:

k = \frac{mv^2}{x^2}

Q: How do we find the compression of the spring?

A: To find the compression of the spring, we need to use the fact that the block's kinetic energy is converted into potential energy stored in the spring. We can set up an equation based on the conservation of energy:

KE = PE

Substituting the equations for kinetic energy and potential energy, we get:

\frac{1}{2}mv^2 = \frac{1}{2}kx^2

We can rearrange the equation to solve for the compression of the spring:

x = \sqrt{\frac{mv^2}{k}}

Q: What is the acceleration of the block as it compresses the spring?

A: The acceleration of the block as it compresses the spring is given by the equation:

a = -\frac{kx}{m}

where k is the spring constant, x is the compression of the spring, and m is the mass of the block.

Q: What is the force on the block as it compresses the spring?

A: The force on the block as it compresses the spring is given by the equation:

F = -kx

where k is the spring constant and x is the compression of the spring.

Q: What is the work done on the block as it compresses the spring?

A: The work done on the block as it compresses the spring is given by the equation:

W = F \cdot x

where F is the force on the block and x is the compression of the spring.

Q: What is the energy of the block as it compresses the spring?

A: The energy of the block as it compresses the spring is given by the equation:

E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2

where m is the mass of the block, v is its velocity, k is the spring constant, and x is the compression of the spring.

Conclusion

In this article, we have answered some of the most frequently asked questions about the physics behind a 9 kg block sliding on a horizontal frictionless surface with a speed of 1.2 m/s, and then hitting a bumper that compresses a spring. We have used the principles of conservation of energy and the behavior of springs to understand the forces involved and the resulting motion of the block.

References

Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.
Young, H. D., & Freedman, R. A. (2012). University Physics. Addison-Wesley.