# 2.5 Motion equations for constant acceleration in one dimension  (Page 6/8)

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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 $2\text{.}{\text{00 m/s}}^{2}$ , how long does it take to travel the 200 m up the ramp? (Such information might be useful to a traffic engineer.)

Strategy

Draw a sketch.

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

Solution

1. Identify the knowns and what we want to solve for. We know that ${v}_{0}=\text{10 m/s}$ ; $a=2\text{.}{\text{00 m/s}}^{2}$ ; and $x=\text{200 m}$ .

2. We need to solve for $t$ . Choose the best equation. $x={x}_{0}+{v}_{0}t+\frac{1}{2}{\text{at}}^{2}$ works best because the only unknown in the equation is the variable $t$ for which we need to solve.

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

$\text{200 m}=\text{0 m}+\left(\text{10}\text{.}\text{0 m/s}\right)t+\frac{1}{2}\left(2\text{.}{\text{00 m/s}}^{2}\right)\phantom{\rule{0.15em}{0ex}}{t}^{2}$

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 $t=t\phantom{\rule{0.25em}{0ex}}\text{s}$ , where $t$ is the magnitude of time and s is the unit. Doing so leaves

$\text{200}=\text{10}t+{t}^{2}\text{.}$

5. Use the quadratic formula to solve for $t$ .

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

${t}^{2}+\text{10}t-\text{200}=0$

This is a quadratic equation of the form

${\text{at}}^{2}+\text{bt}+c=0,$

where the constants are $a=1\text{.}\text{00,}\phantom{\rule{0.25em}{0ex}}b=\text{10}\text{.}\text{0,}\phantom{\rule{0.25em}{0ex}}\text{and}\phantom{\rule{0.25em}{0ex}}c=-\text{200}$ .

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

$t=\frac{-b±\sqrt{{b}^{2}-4\text{ac}}}{2a}\text{.}$

This yields two solutions for $t$ , which are

$t=\text{10}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{and}-\text{20}\text{.}0.$

In this case, then, the time is $t=t$ in seconds, or

$t=\text{10}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{s}\phantom{\rule{0.25em}{0ex}}\text{and}-\text{20}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{s}.$

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,

$t=\text{10}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{s}.$

Discussion

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.

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, $\stackrel{-}{a}=\Delta v/\Delta t$ . 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.

Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
hour glass, pendulum clock, atomic clock?
S.M
tnks
David
how did they solve for "t" after getting 67.6=.5(Voy + 0)t
Find the following for path D in [link] : (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
the topic is kinematics
David
can i get notes of solid state physics
Lohitha
just check the chpt. 13 kinetic theory of matter it's there
David
is acceleration a fundamental unit.
no it is derived
Abdul
no
Nisha
K thanks
David
no it's not its derived
Emmanuel
hi
Hello
Emmanuel
hello
David
Hello Emmanuel
Emmanuel
I'm good
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Emmanuel
how are you too
am cool
Emmanuel
spending time summarizing
Emmanuel
Emmanuel
I am fin
Longwar
ok
hi guys can you teach me how to solve a logarithm?
how about a conceptual framework can you simplify for me? needed please
Villaflor
Hello what happens when electrone stops its rotation around its nucleus if it possible how
Afzal
I think they are constantly moving
Villaflor
yep what is problem you are stuck into context?
S.M
not possible to fix electron position in space,
S.M
Physics
Beatriz
yes of course Villa flor
David
equations of kinematics for constant acceleration
A bottle full of water weighs 45g when full of mercury,it weighs 360g.if the empty bottle weighs 20g.calculate the relative density of mercury and the density of mercury....pls I need help
well You know the density of water is 1000kg/m^3.And formula for density is density=mass/volume Then we must calculate volume of bottle and mass of mercury: Volume of bottle is (45-20)/1000000=1/40000 mass of mercury is:(360-20)/1000 kg density of mercury:(340/1000):1/50000=(340•40000):1000=13600
Sobirjon
the latter is true
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture...take density of water as 1g/cm3 and density of liquid 1.2g/cm3
Lila
plz hu can explain Heisenberg's uncertainty principle
who can help me with my problem about acceleration?
ok
Nicholas
how to solve this... a car is heading north then smoothly made a westward turn during the travel the speed of the car remains constant at 1.5km/h what is the acceleration of the car? the total travel time of the car as it smoothly changed its direction is 15 minutes
Vann
i think the acceleration is 0 since the car does not change its speed unless there are other conditions
Ben
yes I have to agree, the key phrase is, "the speed of the car remains constant...," all other information is not needed to conclude that acceleration remains at 0 during the entire time
Luis
who can help me with a relative density question
Lila
1cm3 sample of tin lead alloy has mass 8.5g.the relative density of tin is 7.3 and that of lead is 11.3.calculate the percentage by weight of tin in the alloy. assuming that there is no change of volume when the metals formed the alloy
Lila
morning, what will happen to the volume of an ice block when heat is added from -200°c to 0°c... Will it volume increase or decrease?
no
Emmanuel
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Kate
BPED..is my course.
Kate
No
Emmanuel
I think it is neither decreases nor increases ,it remains in the same volume because of its crystal structure
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture. take density of water as 1g/cm3 and density of liquid as 1.2g/cm3
Lila
Sorry what does it means"no changes in volume occured"?
Sobirjon
volume can be the amount of space occupied by an object. But when an object does not change in shape it will still occupy the same space. Thats why the volume will still remain the same
Ben
Most soilds expand when heated but if it changes state at 0C it will have less volume. Ice floats because it is less dense ie a larger mass per unit volume.
Richard
how to calculate velocity
v=d/t
Emeka
Villaflor
Villaflor
v=d/t
Nisha
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Mine is good. How about you?
Chase
Hi room of engineers
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Okwethu
hello
akinmeji
Hello
Mishael
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Jerry
hi
Sakhi
hi
H.C
so, what is going on here
akinmeji
Ajayi
good morning ppl
ABDUL
If someone has not studied Mathematics enough yet, should theu study it first then study Phusics or Study Basics of Physics whilst srudying Math as well?
whether u studied maths or not, it is advisable to start from d basics cuz it is essential to know dem
Nuru
yea you are right
wow, you got this w/o knowing math
Thomas
I guess that's it
Thomas
later people
Thomas
mathematics is everywhere
Anand
thanks but dat doesn't mean it is good without maths @Riaz....... Maths is essential in sciences particularly wen it comes to PHYSICS but PHYSICS must be started from the basic which may also help in ur mathematical ability
Nuru
A hydrometer of mass 0.15kg and uniform cross sectional area of 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
16.66 meters?
Darshik
16.71m2
aways
,i have a question of let me give answer
aways
the mass is stretched a distance of 8cm and held what is the potential energy? quick answer
aways
oscillation is a to and fro movement, it can also be referred to as vibration. e.g loaded string, loaded test tube or an hinged door