Motion Graphs and Derivatives Worksheet
a) On a distance vs. time graph (“dt”), what does the slope tell you about?
b) Is the slope constant in the above graph?
c) What does this say about the object’s velocity?
d) What do you know about acceleration then?
ste) Find d f(t) for the graph above. This is known as the 1 derivative.
dt ’ This notation can also be written as f (t) , “f-prime of t”.
2f) Find d f(t) for the graph above. This is known as the 2nd derivative. 2 dt ’’ This notation can also be written as f (t) , “f-double prime of t”.
g) Draw the related velocity vs. time and acceleration vs. time graphs.
a) Completely describe the motion of the object to which the graph
b) Is the slope of this graph constant throughout?
c) What does your answer to part b) tell you about the velocity of this object?
d) If the object’s velocity is changing, what “a-word” is occurring?
e) Is the acceleration + or - ? How do you know?
’ f) Find d f(t) or f (t) for the function graphed above. This is velocity.
2’’ g) Find d f(t) or f (t) for the function graphed above. This is acceleration. 2 dt
h) Do these derivatives tell you something about how the object changes its
motion with time? Explain.
i) Draw the related velocity vs. time and acceleration vs. time graphs. Do
they make sense according to what you know about the motion?
a) Describe completely the motion of the object.
b) Is this realistic? How do you know?
c) Where is there no acceleration? Where is there no velocity?
d) Draw the related velocity vs. time and acceleration vs. time graphs. Do
they make sense according to what you have said about the motion?
For each of these three playground slides, ok one might be a little dangerous, sketch the corresponding “dt”, “vt”, and “at” graphs. For the two curvy slides, be aware that your acceleration is changing in time. This is a new area to explore since last year your acceleration was constant. This should be a little challenging. It will help if you consider how the normal force changes as you slide along the slide. Rough sketches are fine!
5. Find the derivatives of each function. Use your calculus notes to help you
If you can do these without too much trouble, you will be fine.
2 a) y = x+ x - 1
4 2 b) f(t) = 5t- 3t+ 3t – 2 + t + 4
c) f(x)= ln (x+2)
2xd) y = 5e
e) y = 4 sin (2x)
2f) y = cos (x) + x
3g) y = x sin (x)
2h) f(t) = t cos (2t)
23i) f(x) = (x+4)
-2-4j) f(x) = (x +7)
xk) y = xe
2l) y = cos (x)
2 m) f(t) = -4.9t+ 7t + 2
6. Here are four descriptions for the velocity of a hockey puck in the xy plane, all
in meters per second.
2 1) v=-3t+ 4t – 2 and v=6t – 4 xy
2 2) v=-3 and v=-5t+ 6 xy
2 3) v = 2t i – (4t+3)j
4) v = -2t i + 3j
For each description, determine whether the x and y components of the acceleration are constant, and whether the acceleration vector a is constant. In
number 4, if v is in meters per second and t is in seconds, what must be the units of the coefficients –2 and 3?
7. A particle moves so that its position as a function of time in SI units is
2 r = 5 i + 4tj + t k
a) Write an expression for the velocity as a function of time in unit
b) Write an expression for the acceleration as a function of time in
unit vector notation.