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A graph of velocity vs. time of a ship coming into a harbor is shown below. (a) Describe the motion of the ship based on the graph. (b)What would a graph of the ship’s acceleration look like?

Line graph of velocity versus time. The line has three legs. The first leg is flat. The second leg has a negative slope. The third leg also has a negative slope, but the slope is not as negative as the second leg.

(a) The ship moves at constant velocity and then begins to decelerate at a constant rate. At some point, its deceleration rate decreases. It maintains this lower deceleration rate until it stops moving.

(b) A graph of acceleration vs. time would show zero acceleration in the first leg, large and constant negative acceleration in the second leg, and constant negative acceleration.

A line graph of acceleration versus time. There are three legs of the graph. All three legs are flat and straight. The first leg shows constant acceleration of 0. The second leg shows a constant negative acceleration. The third leg shows a constant negative acceleration that is not as negative as the second leg.
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Section summary

  • Graphs of motion can be used to analyze motion.
  • Graphical solutions yield identical solutions to mathematical methods for deriving motion equations.
  • The slope of a graph of displacement x size 12{x} {} vs. time t size 12{t} {} is velocity v size 12{v} {} .
  • The slope of a graph of velocity v size 12{v} {} vs. time t size 12{t} {} graph is acceleration a size 12{a} {} .
  • Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.

Conceptual questions

(a) Explain how you can use the graph of position versus time in [link] to describe the change in velocity over time. Identify (b) the time ( t a , t b , t c , t d , or t e ) at which the instantaneous velocity is greatest, (c) the time at which it is zero, and (d) the time at which it is negative.

Line graph of position versus time with 5 points labeled: a, b, c, d, and e. The slope of the line changes. It begins with a positive slope that decreases over time until around point d, where it is flat. It then has a slightly negative slope.
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(a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in [link] . (b) Identify the time or times ( t a , t b , t c , etc.) at which the instantaneous velocity is greatest. (c) At which times is it zero? (d) At which times is it negative?

Line graph of position over time with 12 points labeled a through l. Line has a negative slope from a to c, where it turns and has a positive slope till point e. It turns again and has a negative slope till point g. The slope then increases again till l, where it flattens out.
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(a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in [link] . (b) Based on the graph, how does acceleration change over time?

Line graph of velocity over time with two points labeled. Point P is at v 1 t 1. Point Q is at v 2 t 2. The line has a positive slope that increases over time.
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(a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in [link] . (b) Identify the time or times ( t a , t b , t c , etc.) at which the acceleration is greatest. (c) At which times is it zero? (d) At which times is it negative?

Line graph of velocity over time with 12 points labeled a through l. The line has a positive slope from a at the origin to d where it slopes downward to e, and then back upward to h. It then slopes back down to point l at v equals 0.

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Consider the velocity vs. time graph of a person in an elevator shown in [link] . Suppose the elevator is initially at rest. It then accelerates for 3 seconds, maintains that velocity for 15 seconds, then decelerates for 5 seconds until it stops. The acceleration for the entire trip is not constant so we cannot use the equations of motion from Motion Equations for Constant Acceleration in One Dimension for the complete trip. (We could, however, use them in the three individual sections where acceleration is a constant.) Sketch graphs of (a) position vs. time and (b) acceleration vs. time for this trip.

Line graph of velocity versus time. Line begins at the origin and has a positive slope until it reaches 3 meters per second at 3 seconds. The slope is then zero until 18 seconds, where it becomes negative until the line reaches a velocity of 0 at 23 seconds.
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A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the cylinder vs. time as it goes up and then down the plane.

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Problems&Exercises

Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.

(a) By taking the slope of the curve in [link] , verify that the velocity of the jet car is 115 m/s at t = 20 s size 12{t="20"`s} {} . (b) By taking the slope of the curve at any point in [link] , verify that the jet car’s acceleration is 5 . 0 m/s 2 size 12{5 "." "0 m/s" rSup { size 8{2} } } {} .

Line graph of position over time. Line has positive slope that increases over time.
Line graph of velocity versus time. Line is straight with a positive slope.

(a) 115 m/s size 12{"115 m/s"} {}

(b) 5 . 0 m/s 2 size 12{5 "." "0 m/s" rSup { size 8{2} } } {}

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Using approximate values, calculate the slope of the curve in [link] to verify that the velocity at t = 10.0 s size 12{t="10"`s} {} is 0.208 m/s. Assume all values are known to 3 significant figures.

Line graph of position versus time. Line is straight with a positive slope.
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Using approximate values, calculate the slope of the curve in [link] to verify that the velocity at t = 30.0 s is 0.238 m/s. Assume all values are known to 3 significant figures.

v = ( 11.7 6.95 ) × 10 3 m ( 40 . 0 – 20 .0 ) s = 238 m/s
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By taking the slope of the curve in [link] , verify that the acceleration is 3 . 2 m /s 2 at t = 10 s size 12{t="10"`s} {} .

Line graph of velocity versus time. Line has a positive slope that decreases over time until the line flattens out.
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Construct the displacement graph for the subway shuttle train as shown in [link] (a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the information on acceleration and velocity given in the examples for this figure.

Line graph of position versus time. Line begins with a slight positive slope. It then kinks to a much greater positive slope.
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(a) Take the slope of the curve in [link] to find the jogger’s velocity at t = 2 . 5 s size 12{t=2 "." 5`s} {} . (b) Repeat at 7.5 s. These values must be consistent with the graph in [link] .

Line graph of position over time. Line begins sloping upward, then kinks back down, then kinks back upward again.
Line graph of velocity over time. Line begins with a positive slope, then kinks downward with a negative slope, then kinks back upward again. It kinks back down again slightly, then back up again, and ends with a slightly less positive slope.
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A graph of v t is shown for a world-class track sprinter in a 100-m race. (See [link] ). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at t = 5 s ? (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?

Line graph of velocity versus time. The line has two legs. The first has a constant positive slope. The second is flat, with a slope of 0.

(a) 6 m/s

(b) 12 m/s

(c) 3 m/s 2 size 12{"3 m/s" rSup { size 8{2} } } {}

(d) 10 s

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[link] shows the displacement graph for a particle for 5 s. Draw the corresponding velocity and acceleration graphs.

Line graph of position versus time. The line has 4 legs. The first leg has a positive slope. The second leg has a negative slope. The third has a slope of 0. The fourth has a positive slope.
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Questions & Answers

what are the calculations of Newton's third law of motiow
Murtala Reply
what is dark matter
apex Reply
(in some cosmological theories) non-luminous material which is postulated to exist in space and which could take either of two forms: weakly interacting particles ( cold dark matter ) or high-energy randomly moving particles created soon after the Big Bang ( hot dark matter ).
Usman
if the mass of a trolley is 0.1kg. calculate the weight of plasticine that is needed to compensate friction. (take g=10m/s and u=0.2)
Declan Reply
what is a galaxy
Maduka Reply
what isflow rate of volume
Abcd Reply
flow rate is the volume of fluid which passes per unit time;
Rev
flow rate or discharge represnts the flow passing in unit volume per unit time
bhat
When two charges q1 and q2 are 6 and 5 coulomb what is ratio of force
Mian Reply
When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?
bimo Reply
is that the answer
nehemiah
why is it proportional
nehemiah Reply
i don't know
Adah
y
nehemiah
what are the relationship between distance and displacement
Usman Reply
They are interchangeable.
Shii
Distance is scalar, displacement is vector because it must involve a direction as well as a magnitude. distance is the measurement of where you are and where you were displacement is a measurement of the change in position
Shii
Thanks a lot
Usman
I'm beginner in physics so I can't reason why v=u+at change to v2=u2+2as and vice versa
Usman
what is kinematics
praveen
kinematics is study of motion without considering the causes of the motion
Theo
The study of motion without considering the cause 0f it
Usman
why electrons close to the nucleus have less energy and why do electrons far from the nucleus have more energy
Theo
thank you frds
praveen
plz what is the third law of thermodynamics
Chidera Reply
third law of thermodynamics states that at 0k the particles will collalse its also known as death of universe it was framed at that time when it waa nt posible to reach 0k but it was proved wrong
bhat
I have not try that experiment but I think it will magnet....
Rev Reply
Hey Rev. it will
Jeff
I do think so, it will
Chidera
yes it will
lasisi
If a magnet is in a pool of water, would it be able to have a magnetic field?.
Stella Reply
yes Stella it would
Jeff
formula for electric current
Chizzy Reply
what is that about pleace
Fokoua
what are you given?
Kudzy
what is current
Fokoua
I=q/t
saifullahi
Current is the flow of electric charge per unit time.
saifullahi
What are semi conductors
saifullahi
materials that allows charge to flow at varying conditions, temperature for instance.
Mokua
these are materials which have electrical conductivity greater than the insulators but less than metal, in these materials energy band Gap is very narrow as compared to insulators
Sunil
materials that allows charge to flow at varying conditions, temperature for instance.
Obasi
wao so awesome
Fokoua
At what point in the oscillation of beam will a body leave it?
Atambiri
what is gravitational force
Adah
what is meant by the term law
Fahd Reply
what is physics
Timilehin Reply
it is branch of science that deal with interaction matter and energy is called physics . and physics is based in experiential observation and quentative measurement.
syed
to briefly understand the concept of physics start with history and a brief history of time by Stephen hawkings is what made me have interest in physics
ayesha
physics is a branch of science which deals with the study of matter, in relation to energy.
Frank
physics is a natural science that involve the study of matter and it's motion through space and time, along with related concept such as energy and force
Shodunke
Physics is the science of natural things. for instance, take classical laws which describe the principles of working of the macro realm and then take the quantum laws which describe the quantum realm. It relates everything in this universe –e. g when you see anything, actually photons penetrate.
Anshuman
why do isotopes of the same group undergo the same chemical reactions ?
Theo
Practice Key Terms 4

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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