Name: ___________________

 

Math 122 Lab 2: Projectile Motion

 

One Dimensional Motion This is the part of section 5.4 that we discussed. Our method of doing two-dimensional projectile motion depends completely on the techniques used for one-dimensional motion. You may want to review or ask questions about that material as you proceed. Recall the following facts.

 

  • Newton’s law: F = ma, where F = force, m = mass and a = acceleration = derivative of velocity
  • Weight equals mass times g, the gravitational constant
  • Gravitational constant g = 9.8 m/s2 and g = 32 ft/s2
  • There are 5280 feet in a mile and 3600 seconds in an hour

 

Two Dimensional Motion  For two-dimensional motion, we separately derive equations for each of the two dimensions. For example, suppose a golf ball is launched from ground level with initial speed 180 mph at angle 10°. Assuming no air resistance, we will compute how high and far the ball goes and graph the flight of the ball. With no air resistance, the only force is gravity which acts vertically. If x represents horizontal motion and y represents vertical motion, then there are no forces in the x direction and gravity works in the y direction. 

 

The initial velocity requires some thought. If we are going to separate x and y equations, we need to separate the horizontal (x) and vertical (y) components of velocity. In the triangle drawn below, the speed (the hypotenuse) is v, the horizontal component is v cosθ and the vertical component is v sinθ.

 

 

Newton’s law for x is x’’ = 0. Initial conditions for x are x(0) = 0 and x’(0) = v cos(10°).

 

Newton’s law for y is like what we did before: y’’ = −32 with y(0) = 0 and y’(0) = v sin(10°). Here, v is the speed of the ball in ft/s. To get this, take 180 mph and multiply by 5280/3600. Now, fill out the rest of this table:

 

 

Vertical                                                           Horizontal

 

y’’ = −32                                                          x’’ = 0

 

y’ =                                                                  x’ =                 

 

y  =                                                                  x =

 

How far does the ball go?

The ball lands when y = 0. Take your equation for y, set it equal to 0 and solve for t. The relevant solution is t ≈ 2.86 seconds. Explain what the other solution represents.

 

 

 

Then substitute t ≈ 2.86 into the x equation and find how far the ball goes. Write it here. Also, since distances in golf are usually given in yards, convert this to yards.

 

 

 

How high does the ball go?

The ball reaches its maximum height when y’ = 0. Take your equation for y’, set it equal to 0 and solve for t. Use this value to find the maximum height.

 

 

 

 

 

 

Graph the flight of the ball.

On your calculator, press MODE and change your Graph option from FUNCTION to PARAMETRIC. Press Enter to save this. Then go to your Y= list. It should have lines for xt1, yt1 and so on.  Enter your equation for x on the line xt1 and enter your equation for y on the line yt1. Press Window and use the values tmin = 0, tmax = 3, xmin = 0, xmax = 750, ymin = 0, ymax = 40. Now graph it!

 

 

 

 

(By the way, real golf shots don’t look like this. We have ignored two forces that are very significant in the flight of a golf ball: air resistance and the Magnus force, which is caused by the spinning of the ball. For a good, there is so much backspin that the Magnus force is as large as gravity and air resistance.)

 

A golf shot going downhill.  Now suppose that the ground is not level, but slopes downward at a constant tilt of 6°. Find an equation of the form y = m x for the ground.

 

 

 

Now, take your equations for x(t) and y(t) (no changes, the forces and initial conditions are the same) and find t such that y = m x. Substitute this value into your x and y equations to determine where the ball lands. The actual (ground) distance that the ball travels is the hypotenuse of the right triangle with sides x and y. Find this distance. Are you surprised how much difference 6° makes?

 

 

 

This part should be written up separately, and can be done in groups of two. There are three different problems below, but you only need to do one. The difficulty is the same, but the sports are different. Choose the one that you and your partner are most interested in. Your report should state the problem, sketch any diagrams that would be helpful, and briefly describe the calculations that you did. State your conclusions clearly. Each group should turn in one report with both group members’ names on it.

This is due on Friday at the beginning of class. Papers turned in later Friday lose 1 point, papers turned in Monday lose 4 points. After that, papers lose 4 points per day late.

 

Tennis Serve

For a tennis serve to be “in” it must clear a 3-foot net that is 39 feet away and then land before the back of the service line 60 feet away. Start with an Andy Roddick serve: the ball starts 10 feet high with an initial speed of 140 mph angled down at 8° below horizontal. How long does the ball take to reach the net? How high is it when it reaches the net? How long does it take to reach the service line? How high is it when it reaches the service line? All in all, is this serve in? Graph this serve (state the values of tmin, tmax, xmin, xmax, ymin and ymax used).

In the following, use angles with one decimal place.

Change the angle (by trial and error) until the ball lands right on the service line.

Change the angle until the ball clips the top of the net.

A good serve must be between these two angles. How much margin of error is there?

 

Football Pass

Assuming that a pass is “on line” it still must be at a catchable height to complete the pass. Assume that the receiver is 30 yards away from the passer, who releases the ball from a height of 6 feet angles up 10° above horizontal with initial speed 80 mph. How long does it take the ball to reach the receiver? How high is the ball when it reaches the receiver? Is this ball catchable? Graph this pass (state the values of tmin, tmax, xmin, xmax, ymin and ymax used).

In the following, use angles with one decimal place.

Change the angle (by trial and error) until the ball reaches a catchable height of 8 feet.

Change the angle until the ball reaches a catchable height of 1 foot.

A completion must be between these two angles. How much margin of error is there?

 

Baseball Pitch

Assuming that a pitch is over the plate it still must be the right height to be a strike. Assume that the range of heights is 24 to 42 inches. Start with a Randy Johnson fastball of 95 mph thrown from a height of 6 feet and angle 2° above the horizontal toward home plate 60 feet away. How long does it take for the ball to reach home plate? How high is the ball when it reaches home plate? Is this a strike? Graph this pitch (state the values of tmin, tmax, xmin, xmax, ymin and ymax used).

In the following, use angles with one decimal place.

Change the angle (by trial and error) until the ball is at the maximum height of 42 inches.

Change the angle until the ball is at the minimum height of 24 inches.

A strike must be between these two angles. How much margin of error is there?