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Solving projectile motion problems

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					                          Solving projectile motion problems
                                           Deepak Chandan
                                          University of Toronto
                                               May 17, 2010

    Solving projectile motion problems can seem like a daunting task, however it is much easier than it appears
and by following a few simple steps one can reduce the likelihood of silly mistakes and work toward’s the
solution in a systematic way.
Step 1. Read the question carefully, at-least two times.
Step 2. Draw a diagram illustrating the main aspects of the question (to your best understanding of the
     question).
Step 3. Indicate the positive and the negative directions on your diagram. This can be done by simply
     drawing a + sign beside your diagram and indicating at the four endpoints as “+” or “−”. This
     important step will ensure that you avoid making silly mistakes with signs in your equations, for
     example - the sign of the acceleration due to gravity. It is a very common mistake, many times
     students are aware that they have decided to take the vertical down as negative at the back of their
     mind, but when solving the equations they will substitute g = 9.8m/s2 and forget the “−” sign. If you
     have the directions laid out in front of you, you are less likely to make sign mistakes.
Step 4. Draw a table, lets call it the “Table Of information (TOI)”. The TOI provides a one stop quick-

                                   Table 1: Sample Table of Information
                                                  X                  Y
                                            i Vx = 4         i Vy = 5
                                     f Vx =i Vx = 4            i Vy =?
                                              ax = 0      ay = −9.8
                                           ∆x = 40             ∆y =?
                                              ∆t =?             ∆t =?

     look at all the information that you currently have and are looking for. In Table 1, the values given
     are an example, and the information that is to be represented in there might change depending on
     what the problem setup is. But the idea is to include everything you know or don’t know and put a
     question mark beside those that you do not know. Some of these unknowns might be those that you
     are interested in, and some you may not - depends on how good you are at using your equations of
     motion. If you are good enough, you can avoid solving for unnecessary unknowns and proceed straight
     to the ones you need. Regardless, the table serves as a good visual tool to remind yourself of what you
     have and what you do not have.
Step 5. Write down the three equations of motions on the paper.
                                                 vf = vi + a∆t                                             (1)
                                                             1
                                                ∆s = vi ∆t + a∆t2                                          (2)
                                                             2
                                             2    2
                                            vf − vi = 2a∆s                                                 (3)


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Figure 1: Example showing the case when the pro- Figure 2: Example showing the case when the pro-
jectile is launched from level ground and returns to jectile is launched horizontally from level ground into
the level ground.                                    open air and consequently undergoes free fall under
                                                     gravity.


Step 6. Now that we have taken the time to systematically present the problem to ourselves, which is going
     to help avoid silly mistakes, we can proceed to solve the problems. How to solve the problems can only
     be best learned by going through a lot of solved examples and solving questions. When you do this
     you will soon recognize that there really isn’t a big diversity to projectile motion problems. Most of
     the problems are just variants of a few types of situations, for example:
        • Something is shot from a level ground and comes back to the level ground some distance away
          (Figure 1). The quantities involved are: i vx = vx , i vy , f vx = vx , f vy , ay , ∆t, ∆x, angle of initial
          launch θ and height h. Some of these will be given to you, some you have to find.
        • Something is shot horizontally from an elevated surface which immediately following the launch
          starts experiencing free fall under gravity while still travelling horizontally forward with a constant
          vx (Figure 2). The quantities involved are: i vx = vx , i vy = 0, f vx = vx , f vy , ay , ∆t, ∆x, θ and
          h. Again some of these quantities will be given to you and some you will have to find.
        • Similar to the first example, but instead of coming back to the ground, the projectile gets em-
          bedded within something or collides with something. Examples of this scenario are, a person
          throwing a ball towards some other person standing on the roof of a building, who catches the
          ball; a firefighter aiming his fire hose at an angle so that the water hits the building at an elevation
          corresponding to where the fire is. In all these cases the projectile ends up at a different elevation,
          so we have a new quantity which depending on the problem we could be asked to solve for - the
          final elevation of the projectile hf .
        • Projectile motion problems of the variety mentioned above could be coupled with other types of
          dynamics problems such as a problem incorporation motion in one direction or a pendulum type
          question or even circular motion (depending on how malicious the person making the test wants
          to be!). In these cases you simply have to solve each problem individually, some result that you
          obtain one part of the problem will be crucial and serve as input to solve the second part of the
          problem.
          An example is the case shown in Figure 4. On the ground level you have water and land. Someone
          on land thrown a stone into the air towards the water. Somethings will be given to you, with the
          final aim of solving for the time it takes for the stone to descend to the bottom of the lake from
          the moment it hits the lake surface (and we’ll ignore drag on the stone in the water so that the
          concept of terminal velocity doesn’t apply). To solve this problem you need to have the velocity
          with which the stone starts to descend at the lake’s surface, but this is just the velocity with


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Figure 3: Two examples of projectile motion problems where the projectile ends up at a different elevation
than the one it started from. (i) Projectile is launched towards someone standing on top of a roof at a
higher elevation, (ii) the projectile starts from an elevation higher than where it ends. The projectile could
be launched horizontally from the higher elevation or at an angle.


           which the stone hits the water at the end of its projectile motion phase. So, you first need to
           solve the projectile motion question, get the final velocity of the stone and use that as the input
           to solve for the motion of the stone in the water (and thus get the time).
     One important thing to realize when solving projectile motion questions is that the motion which is a
     2D motion is simply composed of two 1D motions that are, and this is important, coupled together
     by time. What this means is simply, that, both motions are completed in the same ∆t. This is a very
     important thing to remember and will simplify your life tremendously when solving the equations of
     motion. This links the x equations of motion to the y equations of motion and so when solving the
     equations algebraically, you can eliminate time from a x and a y equation to solve for a variable.




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Figure 4: An example of a coupling between a projectile motion problem and a different dynamics problem.
The second dynamics problem here is 1D motion in the vertical direction. Note: In this problem we are
assuming for the sake of simplicity that the ball’s initial velocity at the water’s surface is vertically down
with the magnitude that is attained at the end of the projectile motion. This is simply because at this time
you people haven’t been introduced to the concept of “drag” in the class and so I did not want to consider
a 2D motion in water (which would involve drag). Furthermore even in this 1D case, there is no drag, and
so the concept of terminal velocity is absent from this example.




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