Chapter Two
Two Dimensional Motion
Problem Solving Strategy
- Draw a picture (mental or on paper)
- List known and unknown variables.
a)Beware -- some data may be extraneous and not necessary to solve the problem.
b)Look for either the starting or ending point being at rest, meaning the velocity has a value of zero.
- Do necessary conversions.
- Choose an equation that can be solved with the known variables.
a)This equation may or may not be the answer you are looking for.
b)It may provide a new variable for use in another equation.
c)This may lead to a succession of equations.
Negative or Positive: You get to select the coordinate system that you use to solve problems. This means you get to decide where the displacement is zero and what direction will be positive or negative. Look at what happens if you have a negative acceleration, such as –9.8 m/s2. Does this mean the object is decelerating (slowing) or does it mean that the object is moving along a negative (perhaps the y) axis? It would depend on the problem. For an object moving on the x axis it would mean decelerating. For an object falling along the y axis, due to gravity, it means the object is accelerating, but in the downward direction. In forces it is easier to use 9.8 m/s2 as a positive number.
Falling Bodies
Displacement: Initial position. We can choose the reference frame / coordinate axis.
If the object ends the problem at the same elevation it started at.
If the object ends the problem at a higher elevation than it started.
If the object ends the problem at a lower elevation than it started.
Velocity, initial: If it is dropped from rest.
If fired upward.
If fired downward.
Velocity, final: At the moment it reaches maximum altitude, right before falling back to earth.
If it hits something on the way up and never reaches max altitude (Rare problem).
On the return trip.
If it lands at the same elevation that the problem began at.
Acceleration of Gravity:
Projectile Motion
Motion in two dimensions happens simultaneously.
- In the x direction the velocity is constant, with no acceleration.
- In the y direction the acceleration of gravity affects upward motion (slowing it down) and causes an increase in downward motion.
- Motion in the x and y direction happen simultaneously, however they can be analyzed separately using vector components.
The following review of variables can be overwhelming to memorize. It is much easier if you think it through or draw a pictorial representation.
Angles:All angles are measured from East. Above the horizon is positive, below negative.
Displacement:
The s is always positive.
If the object ends the problem at the same elevation it started at.
If the object ends the problem at a higher elevation than it started.
If the object ends the problem at a lower elevation than it started.
Velocity, initial:Splits into components, ,
In every problem, we choose to fire it in the positive x direction.
If fired horizontally.
If fired at a positive angle (above the horizon).
If fired at a negative angle (below the horizon).
Velocity, final:Since there is constant velocity in the x direction, initial and final are the same.
At the top of the trajectory
If the object hits something on the way up. Not used in problems very often.
On the return trip.
Resultant from adding vectors and . Has an angle not a + or –.
Strategies that work most of the time.
No time is given. Finding time is the key to all falling body or projectile motion problems
1stsolve for this can be +/- , but is usually minus
2nduse from above to get
3rduse from above to solve for range,
(Alternative: , and the quadratic, followed by )
Time or range x is given. This makes the problem easy since velocity is constant in the x direction.
1stGiven time solve for . Given solve for time.
2ndOnce you have time this is easy, and you don’t need the quadratic.
When an object is dropped or fired horizontally and .
becomes
becomes
These versions are time savers, particularly the last one, since it now no longer requires the quadratic formula.
PROJECTILE MOTION
Time is ruled by gravity and height. Most problems involve yvariables and yequations to solve for time. From time, the distance in the xdirection and the final v can be determined.
holds true for all projectile motion problems.
Equations
a = acceleration
s = displacement
t = time
v = velocity or speed
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