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Wize University Physics Textbook (Master) > Motion in 1/2/3D: Kinematics

Kinematic Graphs

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Kinematic Graphs


Position-time and velocity-time graphs are graphical representations of kinematic data. Valuable information can be gathered by evaluating the slop of these graphs or by determining the area under a curve. These graphs can often be easily converted to data sets but may require some analysis when using the equations of motion.


Here is some useful information we can get from these graphs:
  • The slope of a position-time graph gives velocity.
  • The slope of a velocity-time graph gives acceleration.
  • The area under a velocity-time graph gives the displacement of the object in that amount of time.
  • The area under an acceleration-time graph gives the change in the velocity of the object.
  • A flat line indicates constant velocity or displacement.
  • If the velocity changes suddenly (e.g. a runner changes direction and starts going backward), we model this as short spikes in the acceleration versus time plot.

Watch Out!
Positive velocity and positive acceleration does not necessarily indicate a positive displacement.

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  • In the following diagrams you can find displacement as a function of time for a few different type of motions:

















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Example: Kinematic Graphs


The position-time graph of a rolling marble is shown below. Which statement(s) is/are correct?

A. The marble has a positive velocity at t=0 sec.
B. The marble is slowing down at t=1 sec.
C. The marble is speeding up at t=3 sec.
D. The marble has a positive acceleration at t=2 sec, but its velocity is zero.



Correct statements are:

A. is wrong. Remember that slope of a line tangent to position vs time graph is the particle's velocity. This slope is negative at t=0 which means that velocity is negative at this point.

B. is correct. Close to t=1 s, the tangent line to the curve gets less steep when time passes which means that the size of velocity or speed is getting smaller. So, we have a slowing down motion.

C. is correct. Similar to statement B., here, close to t=3 s, the tangent line is getting more steep which means that speed is rising. So, we have a speeding up motion.

D. is correct because if you look at moments before t=2 sec, at those points, velocity is negative and it gets smaller is size until it reaches zero right at t=2 sec (slop of the tangent line is zero). Since the size of velocity is decreasing, there is a slowing down motion for which velocity and acceleration should have opposite directions and since velocity is negative, acceleration should be positive.

A drone was controlled as follows:
I. At t=0 st=0\ s, the drone was stationary on the surface of the ground (y=0y=0)
II. At t=5 st=5\ s, it started to raise up at a constant speed of 4 m/s4\ m/s
III. After rising for 20  m20\;m, it suddenly changed its vertical velocity to 12 m/s12\ m/sand kept moving upward with this constant velocity
IV. When it reached the height of 100 m100\ mabove the ground it suddenly gained a downward acceleration with a magnitude of 1m/s21 m/s^2while it was moving upward
V. After changing direction at a higher altitude , It got back to the ground level with the same acceleration

Note: This motion is not a free fall because the drone acceleration and velocity are controlled by the electric motors inside it.
At what time did the drone raise its speed to 12 m/s12\ m/s?

Enter your answer in s, do not include units.
A ball moves according to the following v-t graph. (The graph ends at t=110s.)


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Find the final displacement of the object.
Extra Practice
Practice: Drone flight
A drone was controlled as follows:
I. At t=0 st=0\ s, the drone was stationary on the surface of the ground (y=0y=0)
II. At t=5 st=5\ s, it was raised up at a constant speed of 4 m/s4\ m/s
Kinematics
a) Calculate the final velocity in m/sm/s of the car given the following displacement-time graph.
b) What is the maximum velocity, and when does it occur?
c) What is the minimum velocity, and when does it occur?
Kinematics
A ball moves according to the following v-t graph. (The graph ends at t=110s.)
a) Find the final displacement of the object.
b) At t=70st=70s which way is the ball traveling?
Practice: Going for a Jog
Your friend went for a jog, and you are looking at their speed measurements against time recorded to their fitness app. Assume that they ran in a single direction (i.e. the direction of the velocity vector is constant).
Practice: Kinematics plots
Consider the motion of the following object. The position and time at each second from t=0 to t=10s are shown both in a table and in a graph, below.
xt00−51−52−153−20405406457458259010\begin{array}{|r|c|}x & t\\0 & 0\\-5& 1\\-5& 2\\-15 & 3\\-20 & 4\\0&5\\40&6\\45&7\\45&8\\25&9\\0&10 \end{array}
Fastest points on plot
Which of the following statements are true at the times (1s, 2s, 3s, 4s)?
What is the total Δy\Delta y from x = 2 to 8?
Sketch the derivative of the following graph
Linearise this!
Given this equation
X5 = 23Y3Z7X^5\ =\ \sqrt{\frac{23Y^3}{Z^7}}
Which of the following plots would be linear?
Proportionality
Given the following equation
X=6YZ4X = \frac{6 \sqrt{Y}}{Z^4}
Which plots will give you a straight line?
Proportionality sketches
Match the following plot sketches with the proportionalities.
Drop the ball
So let's consider a ball that is dropped, starting above the ground, bouncing off the ground and reaching half the original height. And then have the ball fall back to the ground again. Sketch the position/velocity/acceleration graphs.
Vanessa is making her way downtown, moving quite fast. She's racing Adam who's in a big yellow taxi. Their velocities are shown below
Over the entire race, who travelled the longest distance? Who has the greater displacement?
Draw out plot
A driver is traveling at 2.0m/s, and then slows down to a stop at a constant rate in 2.0s. Then he starts accelerating again at 1.5m/s2 for 3.0s.
Draw out the velocity/acceleration-time graphs for the motion.
Instantaneous and average velocity
With the following position-time plot, you can get the average and instanteous velocities. Are the following statements true or false?
Integral of v-t graph
Given the following velocity-time graph
Rank, from lowest to greatest, the velocities at each of the points (v⃗a,v⃗b,v⃗c,v⃗d\vec{v}_a, \vec{v}_b, \vec{v}_c, \vec{v}_d). Then rank, from lowest to greatest, the speeds at each point (va,vb,vc,vdv_a, v_b, v_c, v_d).
Sketch the position-time and acceleration-time graphs for the following motion.
Velocity from acceleration-time plot
Given the following acceleration-time plot, which velocity-time plot is possible?
Fastest points on plot
Which of the following statements of the speeds at the times (1s, 2s, 3s, 4s) is correct?
Linearise
Which plot would give a linear line?
X=12Y3Z2X=\frac{12Y^3}{Z^2}
Practice: Kinematic Graph
draw velocity-time graph corresponding to given position-time graph.
Practice: Acceleration
A drone was controlled as follows:
I. At t=0 st=0\ s, the drone was stationary on the surface of the ground (y=0y=0)
II. At t=5 st=5\ s, it started to raise up at a constant speed of 4 m/s4\ m/s
Calculate the final velocity (in m/sm/s) of the car given the following displacement-time graph. Find the maximum and minimum velocities and when they occur.
A ball moves according to the following v-t graph.
a) Find the final displacement of the object.
b) At t=70st=70swhich way is the ball traveling?