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UNIFORMLY ACCELERATED MOTION IN 1DMs. Mikaela Fudolig
Physics 71
2nd sem 12-13
Deriving the kinematic equations
Start with the definitions
dva
dtdx
vdt
Note: ACCELERATION and VELOCITY are VECTORS. However, in 1D, the arrows above the letters are usually removed if the context is clear.
Deriving the Kinematic Equations
Assuming that the acceleration is CONSTANT, we get the following:
0
20 0
( )
1( )
2
v t v at
x t x v t at
NOTE:
v0 is the velocity (not speed) at time t=0
x0 is the position at time t=0
Deriving the Kinematic Equations
From these two, we can derive two more:
2 20
0
( ) 2
( )
2
v t v a x
v t vx
t
0
:
( )
NOTE
x x t x
Solving Kinematic Problems
Define your AXIS. Define the POSITIVE and NEGATIVE
directions. Identify the GIVEN. Identify the UNKNOWNS. Which kinematic equation/s will give
you the solution for the unknown? Hint: look for the equation/s with the
least number of unknowns.
Solving Kinematic Problems
If the quadratic equation results in two values of t, check which is the more reasonable of the two values based on the problem statement.
Example 1
A stone fell from the top of a building 50.0m above ground. Assume that “UPWARDS” is positive. Upon reaching the ground: what is the stone’s displacement?
what is the distance traveled?
Example 2
A car is initially moving at a speed of 15m/s, and it is accelerating at a rate of 2m/s2. What is its speed after 5s? (25m/s)
Example 3
A car is initially moving at a speed of 15m/s and is moving at a constant acceleration. Fifteen seconds later, his speed is 40m/s. What is its acceleration (1.67m/s2)?
Example 4
A car is initially moving at a speed of 15m/s, and it is accelerating at a rate of 2m/s2. How far is the car from its initial position after 2 seconds? (34m)
Example 5
You are driving a car at a constant speed of 15m/s. You put down the brakes, slowing down the car at a rate of 5m/s2. How far did the car move before coming to a complete stop? (22.5m)
Freefall
An object falling under the sole influence of gravity. Neglect air resistance. Assume that the acceleration due to gravity is
constant. An object in freefall has an acceleration of
2ˆ9.81grav
ma j
s
Assumption:
The vector j points upward, away from the ground.
The constant g
To be more consistent with later lectures, we assume that g refers to the MAGNITUDE of the acceleration due to gravity:
29.81
mg
s ˆ
grava gj
Freefall
An object in “freefall” does not have to be falling. Objects thrown up into the air also
undergo freefall, as long as they are only acted upon by gravity.
Under this assumption, objects in freefall move in the same manner regardless of mass, size, etc.
Freefall
In 1D, we usually omit the unit vectors. SIGNS are VERY important.
The acceleration due to gravity is usually written simply as
29.81
ma g
s
Freefall
At ALL points along its motion, an object in freefall has the same acceleration!
Example 1
You throw an object downwards. What is the object’s acceleration (magnitude and direction): Right after being released? Somewhere between its initial
position and the ground? Right before reaching the ground?
Up and Down Motion (1D)
An important aspect about up-and-down motion is:
At the TOP of its flight, the VELOCITY of the
object is ZERO!
Exercise 1
You throw an object upwards. What is the object’s acceleration (magnitude and direction): Right after being released? Somewhere between its initial position and
its maximum height? At its maximum height? Somewhere between its maximum height
and the ground? Right before reaching the ground?
Example 2
You throw an object downwards with an initial speed v0. At t=tf, the object reaches the ground. Draw the a vs. t diagram of the object. Draw the v vs. t diagram of the object. Draw the y vs. t diagram of the object.
Exercise 2
You throw an object upwards with an initial speed v0. At t=tf, the object reaches the ground. Draw the a vs. t diagram of the object. Draw the v vs. t diagram of the object. Draw the y vs. t diagram of the object.
Solving Problems in Freefall (1D)
Freefall is also uniformly accelerated motion.
Just replace a with –g= –9.81m/s2 in the kinematic equations shown earlier. Note that g>0, a=–g<0
It may be useful to use the fact that the velocity is zero at the maximum height.
Solving Problems in Freefall (1D)
Then we have:
0
20 0
2 20
0
1
2
2
2
v v gt
y y v t gt
v v g y
v vy
t
Solving Problems in Freefall (1D)
Note that SIGNS are important. If you are given the speed and the
direction of motion, MAKE SURE that what you input in the equations is the VELOCITY, NOT the speed.
What is the difference between an object thrown upward and an object thrown downward, each with the same initial speed?
Solving Problems in Freefall (1D)
1. DRAW the situation!!!2. What are the GIVEN quantities?
If the given quantities are speeds, remember to convert them to velocities.
3. What are the UNKNOWNS?4. Is it necessary to use the fact that the
velocity at the maximum height is zero?5. Solve the necessary kinematic
equations.
Extra hints: Freefall in 1D
1. Look for hidden clues. “dropped” = initial velocity is zero “top of its flight” / “maximum height” =
velocity is zero at this point “speed” vs. “velocity”
2. Read the problem carefully. Draw the situation BEFORE attempting to
solve.
Example 3
An object is thrown downwards with an initial speed vi from a height h above the ground. What is its velocity upon reaching the
ground? How long will it take to reach the ground?
Exercise 3
An object is thrown upwards with an initial speed vi from a height h above the ground. What is its velocity upon reaching the
ground? How long will it take to reach the ground?
Example 4
An object is thrown upwards from the top of a building a distance ybuilding above the ground. It reaches its maximum height a distance hmax from the top of the building. What is its velocity upon reaching the
ground? With what speed was the object thrown? What is the object’s velocity upon reaching
the ground? What is the total time of flight?
Example 4
An object is thrown upwards from the top of a building a distance ybuilding above the ground. It reaches its maximum height a distance hmax from the top of the building. With what speed was the object thrown? What is its velocity upon reaching the
ground? What is the object’s velocity upon reaching
the ground? What is the total time of flight?
Example 5
An object is thrown upwards from the top of a building a distance ybuilding above the ground. It reaches its maximum height a time tmax later. With what speed was the object thrown? What is the object’s velocity upon reaching
the ground? How long did it take the object to complete
its flight? What is the maximum height reached by
the object?
Example 5
An object is thrown upwards from the top of a building a distance ybuilding above the ground. It reaches its maximum height a time tmax later. With what speed was the object thrown? What is the object’s velocity upon reaching
the ground? How long did it take the object to complete
its flight? What is the maximum height reached by
the object?
Summary
Kinematic equations for uniformly accelerated motion can be derived from the definitions of the acceleration and velocity.
Signs are important. Freefall is uniformly accelerated motion.
a = –g = –9.81m/s2
Maximum height
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