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Calculating time: a car merges into traffic

Suppose a car merges into freeway traffic on a 200-m-long ramp. If its initial velocity is 10.0 m/s and it accelerates at 2 . 00 m/s 2 size 12{2 "." "00 m/s" rSup { size 8{2} } } {} , how long does it take to travel the 200 m up the ramp? (Such information might be useful to a traffic engineer.)


Draw a sketch.

A line segment with ends labeled x subs zero equals zero and x = two hundred. Above the line segment, the equation t equals question mark indicates that time is unknown. Three vectors, all pointing in the direction of x equals 200, represent the other knowns and unknowns. They are labeled v sub zero equals ten point zero meters per second, v equals question mark, and a equals two point zero zero meters per second squared.

We are asked to solve for the time t size 12{t} {} . As before, we identify the known quantities in order to choose a convenient physical relationship (that is, an equation with one unknown, t size 12{t} {} ).


1. Identify the knowns and what we want to solve for. We know that v 0 = 10 m/s size 12{v rSub { size 8{0} } ="10 m/s"} {} ; a = 2 . 00 m/s 2 size 12{a=2 "." "00 m/s" rSup { size 8{2} } } {} ; and x = 200 m size 12{x="200 m"} {} .

2. We need to solve for t size 12{t} {} . Choose the best equation. x = x 0 + v 0 t + 1 2 at 2 works best because the only unknown in the equation is the variable t size 12{t} {} for which we need to solve.

3. We will need to rearrange the equation to solve for t size 12{t} {} . In this case, it will be easier to plug in the knowns first.

200 m = 0 m + 10 . 0 m/s t + 1 2 2 . 00 m/s 2 t 2 size 12{"200 m"="0 m"+ left ("10" "." "0 m/s" right )t+ { {1} over {2} } left (2 "." "00 m/s" rSup { size 8{2} } right )t rSup { size 8{2} } } {}

4. Simplify the equation. The units of meters (m) cancel because they are in each term. We can get the units of seconds (s) to cancel by taking t = t s size 12{t=t" s"} {} , where t size 12{t} {} is the magnitude of time and s is the unit. Doing so leaves

200 = 10 t + t 2 . size 12{"200"="10"t+t rSup { size 8{2} } } {}

5. Use the quadratic formula to solve for t size 12{t} {} .

(a) Rearrange the equation to get 0 on one side of the equation.

t 2 + 10 t 200 = 0 size 12{t rSup { size 8{2} } +"10"t - "200"=0} {}

This is a quadratic equation of the form

at 2 + bt + c = 0 ,

where the constants are a = 1 . 00, b = 10 . 0, and c = 200 size 12{a=1 "." "00,"`b="10" "." "0,"`"and"`c= - "200"} {} .

(b) Its solutions are given by the quadratic formula:

t = b ± b 2 4 ac 2 a .

This yields two solutions for t size 12{t} {} , which are

t = 10 . 0 and 20 . 0 . size 12{t="10" "." 0``"and"`` - "20" "." 0} {}

In this case, then, the time is t = t size 12{t=t} {} in seconds, or

t = 10 . 0 s and 20 . 0 s . size 12{t="10" "." 0``s`"and" - "20" "." 0`s} {}

A negative value for time is unreasonable, since it would mean that the event happened 20 s before the motion began. We can discard that solution. Thus,

t = 10 . 0 s . size 12{t="10" "." 0`s} {}


Whenever an equation contains an unknown squared, there will be two solutions. In some problems both solutions are meaningful, but in others, such as the above, only one solution is reasonable. The 10.0 s answer seems reasonable for a typical freeway on-ramp.

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With the basics of kinematics established, we can go on to many other interesting examples and applications. In the process of developing kinematics, we have also glimpsed a general approach to problem solving that produces both correct answers and insights into physical relationships. Problem-Solving Basics discusses problem-solving basics and outlines an approach that will help you succeed in this invaluable task.

Making connections: take-home experiment—breaking news

We have been using SI units of meters per second squared to describe some examples of acceleration or deceleration of cars, runners, and trains. To achieve a better feel for these numbers, one can measure the braking deceleration of a car doing a slow (and safe) stop. Recall that, for average acceleration, a - = Δ v / Δ t size 12{ { bar {a}}=Δv/Δt} {} . While traveling in a car, slowly apply the brakes as you come up to a stop sign. Have a passenger note the initial speed in miles per hour and the time taken (in seconds) to stop. From this, calculate the deceleration in miles per hour per second. Convert this to meters per second squared and compare with other decelerations mentioned in this chapter. Calculate the distance traveled in braking.

Questions & Answers

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