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A 4.0 kg block is moving at 5.0 m/s along a horizontal frictionless surface toward and ideal spring that is attached to a wall , After the block collides with the spring, the spring is compressed a maximum distance of 0.68m . what is the speed of the block when the spring is compressed to only one-half of the maximum distance?

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ajeigbeibraheem

A 4.0 kg block is moving at 5.0 m/s along a horizontal frictionless surface toward an ideal spring that is attached to a wall, the maximum speed of the block when the spring is compressed to one-half of the maximum distance is 4.33 m/s

From the conservation of energy; the kinetic energy of the mass is equal to the work done on the spring.

i.e.

[tex]\mathbf{\dfrac{1}{2} mv^2 = \dfrac{1}{2}kx^2_{max}}[/tex]

Given that:

  • the mass of the block = 4.0 kg
  • the speed at which it is moving = 5.0 m/s
  • compression of the spring = 0.68 m

∴

From the equation above, multiplying both sides with 2, we have:

[tex]\mathbf{mv^2 =kx^2_{max}}[/tex]

Making (k) the subject of the formula;

[tex]\mathbf{k = \dfrac{mv^2}{x^2_{max}}}[/tex]

[tex]\mathbf{k = \dfrac{4 \times 5^2}{0.68^2}}[/tex]

k = 216.26 N/m

However, when compressed to one-half of the maximum distance; the speed is computed as follows:

x = 0.68/2 = 0.34 m

∴

[tex]\mathbf{\dfrac{1}{2}mv_o^2 - \dfrac{1}{2}mv^2 = \dfrac{1}{2}kx^2}[/tex]

[tex]\mathbf{m(v_o^2 -v^2) =kx^2}[/tex]

[tex]\mathbf{(v_o^2 -v^2) =\dfrac{kx^2}{m}}[/tex]

[tex]\mathbf{(5^2 -v^2) =\dfrac{216.26 \times 0.34^2}{4.0}}[/tex]

25 - v² = 6.25

25 -6.25 = v²

v² = 18.75

[tex]\mathbf{ v= \sqrt{18.75 }}[/tex]

v = 4.33 m/s

Therefore, we can conclude that the speed of the block when the spring is compressed to only one-half of the maximum distance is 4.33 m/s

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