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# When you drive behind another car, what is the minimum distance you should keep between the cars to avoid a rear-end collision if the other car were to suddenly stop (it hits, say, a stationary truck)? Some drivers use a " $2$ second rule" while others use a "$3$ second rule." To apply such rules, pick out an object such as a tree alongside the road. When the front car passes it, begin to count off seconds. For the first rule, you want to pass that object at a count of $2 \mathrm{~s}$, and for the second rule, $3 \mathrm{~s}$. For the $2 \mathrm{~s}$ rule, what is the resulting car-car separation at a speed of $(a)$ $15.6 \mathrm{~m} / \mathrm{s}$ ($35 \mathrm{~m} / \mathrm{s}$), slow and $(b)$ $31.3 \mathrm{~m} / \mathrm{s}$ $(70 \mathrm{mi} / \mathrm{h}$), fast? For the $3 \mathrm{~s}$ rule, what is the car-car separation at a speed of $(c)$ $15.6 \mathrm{~m} / \mathrm{s}$ and $(d)$ $31.3 \mathrm{~m} / \mathrm{s}$ ? To check if the results give safe trailing distances, find the stopping distance required of you at those initial speeds. Assume that your car's braking acceleration is $-8.50 \mathrm{~m} / \mathrm{s}^2$ and your reaction time to apply the brake upon seeing the danger is $0.750 \mathrm{~s}$. What is your stopping distance at a speed of $(e)$ $15.6 \mathrm{~m} / \mathrm{s}$ and $(f)$ $31.3 \mathrm{~m} / \mathrm{s}$ ? $(g)$ For which is the $2 \mathrm{~s}$ rule adequate? $(h)$ For which is the $3 \mathrm{~s}$ rule adequate ?

Solution

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Givens

• $v_1=15.6\ \frac{\text{m}}{\text{s}}$ - the slower speed
• $v_2=31.3\ \frac{\text{m}}{\text{s}}$ - the faster speed
• $\tau_1=2\ \text{s}$ - the two second rule
• $\tau_2=3\ \text{s}$ - the three second rule
• $a=8.5\ \frac{\text{m}}{\text{s}^2}$ - the deceleration
• $\Delta t=0.75\ \text{s}$ - the reaction time

Required

• determine which of the rules is appropriate in which situation

How can we use the known expressions for the speed and the distance travelled by the car in order to solve this problem?

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