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Question

# If the truck brakes and the crate slides to the right relative to the truck, the horizontal acceleration of the crate is given by $\overline{s}=-g \mu_{k},$ where g is the acceleration of gravity, $\mu_k=0.87.$ is the kinetic friction coefficient, and s is the position of the crate relative to a coordinate system attached to the ground (rather than the truck). Assuming that the crate slides without hitting the right end of the truck bed, determine the time it takes to stop if its velocity at the start of the sliding motion is $v_0 = 55 mph.$

Solution

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Find the time $t$ it takes for the truck to stop after applying brakes.

Convert $v_o=55\text{mph}$ to $\dfrac{\text{ft}}{\text{s}}$.

\begin{align*} v_o&=55\text{mph}\cdot\dfrac{5280\text{ft}}{1\text{mi}}\cdot\dfrac{1\text{hr}}{3600\text{s}}=80.67\dfrac{\text{ft}}{\text{s}} \end{align*}

In solving for $t$, we have several equations derived in $\textbf{Kinematics}$. To easily identify which equation must be used, analyze the given data.

In this problem, we are given $v_o$ as our initial velocity, $v$ as our final velocity, $a$ as our acceleration which is expressed as $-\mu_kg$ ($\mu_k$ is 0.87), and $t$ as our missing quantity. With that, we can use $v=v_o+at$.

Thus,

\begin{align*} 0&=80.67+(-0.87\cdot32.2)\cdot t \\ t&=\boxed{\textcolor{#4257b2}{2.88\text{s}}} \end{align*}

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