Question

do not include units in your answer. each answer that is not a whole number should be rounded to one decimal place (the tenths place). 6. a 12 - kilogram crate travels along a horizontal frictionless surface at a constant speed of 6.0 meters per second. the crate compresses a spring a distance of 0.40 meter and comes to a complete stop. determine the kinetic energy of the crate. 7. a 12 - kilogram crate travels along a horizontal frictionless surface at a constant speed of 6.0 meters per second. the crate compresses a spring a distance of 0.40 meter and comes to a complete stop. determine the spring constant of the spring.

Explanation:

Step1: Recall kinetic energy formula

The kinetic energy formula is $KE = \frac{1}{2}mv^2$, where $m$ is mass, $v$ is speed.

Step2: Substitute given values

$m=12$, $v=6.0$. Substitute into formula:
$KE = \frac{1}{2} \times 12 \times (6.0)^2$

Step3: Calculate the kinetic energy

First compute $(6.0)^2=36$, then $\frac{1}{2} \times 12 = 6$, finally $6 \times 36 = 216$
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Step4: Relate KE to spring potential energy

Kinetic energy converts to spring potential energy: $KE = \frac{1}{2}kx^2$, where $k$ is spring constant, $x$ is compression distance.

Step5: Rearrange to solve for $k$

$k = \frac{2KE}{x^2}$

Step6: Substitute values for $k$

$KE=216$, $x=0.40$. Substitute into formula:
$k = \frac{2 \times 216}{(0.40)^2}$

Step7: Calculate the spring constant

First compute $(0.40)^2=0.16$, $2 \times 216=432$, then $\frac{432}{0.16}=2700$

Answer:

1. 216
2. 2700