Class Exercise September 13th
Please not there was a typographical error on the sheet you used in class. The vertical velocity component is 20.0 m/s not 2.0 m/s!!!
Problem 17: Whammo the magnificant is launched from a cannon and is to land in a net 10.0 m below the launch point. If Whammo's initial velocity components are 20.0 m/s upward and 10.0 m/s horizontally, how long is he in the air? Where should the net be placed? Does he miss the wall?
1A: Choose a coordinate system for solving this problem. Choose the direction of the coordinate axes, the position of the origin, and the name of the coordinate(s). Write out your choices below, using complete sentences.
1B: Draw a diagram showing your coordinates. Mark on your diagram all of the relevant points described in the problem statement.
2: Write down all of the given information using relevant algebraic symbols (e.g. vx and also write down a symbol for any unknowns you are asked to find.
3: Write down any relevant mathematical relations that will help you solve the problem. Do not write down relations that may be correct but are not useful for this problem If you change your mind, just draw a thin line through the irrelevant information.
4: Solve for the time he is in the air. Hint: do you know the value of one of the coordinates when he lands?
5: Solve for the position of the net. Hint: you have just found the time from launch to net.)
6: How can you determine whether he misses the wall? Write out your PLAN for solving this part of the problem before you start calculating.
Two projectile paths pass through a given goal.
Class exercise for September 15th
You want to throw a ball so as to reach a friend 12 m away. If the desired target (your friend’s hand) is at the same height from which you launch the ball, and you can throw at 13 m/s, at what angle should you throw the ball?
I will model the ball as a projectile moving in a plane. I will neglect air resistance.
Draw a diagram, choose a coordinate system and write all the given information in terms of the coordinates. Don’t forget to assign a symbol to the unknown quantity you want to find.
I will put the origin at the thrower’s hand, the x-axis horizontal toward the friend, and the y-axis vertically upward. The angle I am trying to find is measured from the horizontal and is called q . The velocity components at launch are
vyi = v0sinq , vxi = v0cosq , where v0 = 13 m/s. The coordinates are:
xi = 0, yi = 0, xf = d = 12 m, yf = 0,
Next decide how you will solve the problem, and write down the equations you plan to use.
I will solve for the time that the projectile is in the air. The answer will depend on q . Then I will put the time in the x-equation and solve for q .
yf = 0 = yi+ +vyit+1 ayt2 = 0+ v0sinq t-1 g t2
xf = d = xi+ +vxit = 0+ v0cosq t
SOLVE Solve the problem.
t = d/ v0cosq ; v0sinq -1 gd /v0cosq = 0;
2cosq sinq = sin2q = gd/ v02=(10 m/s2)(12 m)/(13 m/s)2=0.71 (no units - they all cancel)
Thus 2q = 45° or 135°
q=22.5° or 67.5°
ANALYZE Discuss your result
There are two possible angles. The ball will take longer to reach my friend if I use the greater angle, that’s a better choice for us because neither of us catch well, and we will have more time to judge how to catch the ball.