t
v
i
v
f
Velocity vs. Time
Deriving the 3 Kinematics Equations
In this activity, you will be using concepts we have already covered in class to derive three new
kinematics
equations
for constant acceleration
. You will use the graph below to work through this
derivation.
1.
Describe the motion of an object whose velocity vs. time
graph is shown to the right.
2.
Is the acceleration constant over the time interva
l
,
t?
How do you know?
3.
Solve for the acceleration in terms of v
f
, v
i
, and t. Then rearrange the equation to solve for v
f
.
This is equation # 1. Box it and write a big
next to it.
4.
Using the graph, write an expression for the displacement
(
⃗
)
of the object during the time interval
,
t. Your
expression should be in terms of v
f
, v
i
, and t.
5.
This equation is a little ugly. We can make it prettier by substituting for
v
f
. Use e
quation
(which
should look like v
f
= ________) to substitute for v
f
.
This is equation # 2. Box it and write a big next to it.
6.
Now that we have these
two
equations, it might be nice if we could have one without time
.
Solve
for t.
7.
** Warning: this step is tough. Try it if you’d like. Otherwise, wait for me to go over it with the class.**
Substitute your expression for t into .
See if you can simplify and solve for
.
This is equation # 3. Box it and write a big next to it.
Kinematics Equations
Strategy steps courtesy of www.physicsclassroom.com
What do I know?
v
i
=
v
f
=
t =
a =
∆
x =
Kinematics
Problem Solving Strategy and Practice
Strategy
1.
Construct an informative diagram of the physical situation.
2.
Identify and list the
following:
giv
en information in variable form
unkno
wn information in variable form
equation that will be used to determine unknown information from
the givens
3.
Substitute known values into the equation and use appropriate algebraic ste
ps to solve for the unknown
information.
4.
Check your answer to insure that it is reasonable and mathematically correct.
Practice
1.
John
is riding along in his new car when he gets stuck behind a dump truck. After some time he
gets a
p
assing lane
and hits the gas, speeding
up from
17.8
m/s to 3
2
.6 m/s in 4.6 seconds.
a.
What was his average acceleration during this time?
b.
How far did he travel while making this pass?
2.
Uh oh. John
spot
s
a police cruiser coming around the corner and hit
s
the brakes, slowing to
27 m/s. His
car travels 100 meters while braking.
How much time did this take?
You may have to solve for an intermediary unknown before
you can solve for
time.
What do I know?
v
i
=
v
f
=
t =
a =
∆
砠㴠
3.
A car is initially traveling due north at 23 m/s.
a.
Find the velocity of the car after 4 s if its acceleration is
2 m/s
2
due north.
b.
Find the velocity of the
car after 4 s
if its acceleration is instead
2 m/s
2
due south.
4.
You are driving along the street at the speed limit (35mph) and 50 meters before reaching a traffic light
you notice it turn yellow. You accelerate to make the traffic light within th
e 3 seconds it takes for it to
turn red. What is your speed as you cross the intersection? Assume that the acceleration is constant.
(1.0 mile = 1609 m)
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