How Far Does a Block Compress a Spring on a Frictionless Surface?

[ice2morrow]

Homework Statement

A 4.0 kg block slides along a frictionless horizontal surface at 5.0 m/s. It runs into and compresses a spring with a spring constant of 200 N/m. How far does it compress the spring before it stops and changes direction?

Homework Equations

P.E. = 0.5 kx^2
K.E. = 0.5 mv^2

The Attempt at a Solution

x = sqrt (mv^2/k) = 0.707 m?

 

[Chewy0087]

that's right.

 

[kerimek]

I would first like to clarify the given information and assumptions. It is stated that the block is sliding on a frictionless surface, which means there is no external force acting on the block to slow it down. Therefore, the block will continue to move with a constant velocity of 5.0 m/s until it encounters the spring. Additionally, the spring constant of 200 N/m implies that the spring is a linear spring, meaning that the restoring force is directly proportional to the displacement of the spring from its equilibrium position.

To solve this problem, we can use the conservation of energy principle. Initially, the block has only kinetic energy, given by 0.5 * 4.0 kg * (5.0 m/s)^2 = 50 J. When the block compresses the spring, this kinetic energy is converted into potential energy stored in the spring. Using the equation for potential energy of a spring, we can calculate the maximum compression distance x:

PE = KE
0.5 * k * x^2 = 50 J
x = sqrt(100 J/200 N/m) = 0.707 m

Therefore, the block will compress the spring by 0.707 m before it stops and changes direction. It is important to note that this is the maximum compression distance, as the block will continue to oscillate back and forth around the equilibrium position of the spring until all of its kinetic energy is converted into potential energy and vice versa.

In conclusion, kinematics and springs are important concepts in physics that allow us to understand the motion of objects and the behavior of elastic materials. By applying the principles of conservation of energy and understanding the properties of springs, we can accurately predict the motion of the block in this scenario.