Lab #3: 2-Dimensional Kinematics Projectile Motion

Lab #3: 2-Dimensional Kinematics Projectile Motion

A medieval trebuchet by Kolderer, c1507

Reading Assignment:

Chapter 3, Sections 3-1 through 3-6 Chapter 4, Sections 4-1 through 4-6

Introduction:

In medieval days, people had a very practical knowledge of projectile motion. They may not have understood the exact trajectory that a projectile would take, but by practice they could place a projectile on a target consistently from a distance of well over 200 yards. During a long siege of a castle, it was not uncommon to hurl bodies of animals (and yes, captives) back into the besieged castle's water supply (an early form of biological warfare). Similarly, a modern day hunter does not need to know the actual path that a bullet takes to a target in order to hit the target. A sharpshooter, however, does know the path and can make adjustments in the aiming in order to hit a target at many different ranges. In this lab, you will become a sharpshooter of sorts. You will use the equations of motion to predict the path of a projectile and hit a target.

Neglecting frictional forces, such as air resistance, an object projected from a launcher undergoes a motion that is the simple vector combination of uniform velocity in the horizontal direction and uniform acceleration in the vertical direction. For a projectile launched with a speed, v(0), at an angle with respect to the positive x axis, it can be shown that the trajectory caused by such a combination predicts a parabolic shape. The following kinematic equations describe this motion:

Horizontal Motion:

Vertical Motion:

x(t) = x(0) + vx (0)t Eq 1 vx (t ) = vx (0)

y(t) = y(0) + vy (0)t + 1 2 ayt 2

Eq 2

vy (t) = vy (0) + ayt

Eq 3

vy (t )2 = vy (0)2 + 2ay ( y(t) - y(0)) Eq 4

Lab#3 ? 2D Kinematics

Where vy(0) and vx(0) are the initial vertical and horizontal components of the velocity respectively.

Notice that Equations 1 and 2 have a common variable, t. Equation 1 predicts the x coordinate in terms of the parameter t, Equation 2 predicts the y coordinate in terms of the parameter, t. By combining these two equations, the dependency upon the parameter, t, can be eliminated. Simply solving Equation 1 for t and substituting it into Equation 2 results in the following:

y(x)

=

y(0)

+

vy

(0)

x vx (0)

+

1 2

a

y

x 2 vx (0)

Where: Which simplifies to:

x = x(t) - x(0)

y(x)

=

y(0)

+

vy (0) vx (0)

x

+

ay

2(vx (0))2

(x)2

Eq

5

Furthermore, the components of the velocity can be written in terms of the original launch velocity as:

vx (0) = v(0) cos

vy (0) = v(0)sin

These components, when combined with Equation 5 yield an equation for y(x) determined completely by v(0) and (the initial launch speed and angle):

y(x)

=

y(0)

+

(tan )x

+

ay

2(v(0) cos )2

(x)2

Eq 6

Notice that Equations 5 and 6 describe the position of the object but they do not say when (at what time) the object has any particular position.

Also, notice that the relationship between y and t in Equation 2 is quadratic (parabolic) in t because the values for ay,

vy(0), and y(0) are constant. Similarly, in Equations 5 and 6, vertical position, y, as a function of horizontal position, x, is quadratic (parabolic) in ? x because the values for ay , vy(0) , and vx(t) are also constant. The equation for y(x) represents the trajectory of the projectile.

If ay, x(t), x(0), , y and y(0) are known, then it should be possible to determine the speed at which the projectile was launched.

Note: The famo us "Range Equation" for projectile motion is a special case of the derivation described above. It can only be used when a projectile starts and lands at exactly the same vertical height. It also defines a coordinate axis for the trajectory such that x(0) = 0 and y(0) = 0.

As an exercise, plot Equation 6 as y vs. x for a variety of realistic values for , v(0), and y(0). What determines the shape of the curve, the x position of the maximum, and the height of the curve (y position of the maximum)? Experimenting with the mathematics of a trajectory can yield tremendous insight into projectile motion.

Lab#3 ? 2D Kinematics

Lab #3: 2-Dimensional Kinematics

Goals:

? Resolve velocity vectors into components. ? Determine the muzzle velocity of a projectile launcher. ? Predict the range of a projectile. ? Use ExcelTM to analyze the motion of a projectile.

Equipment:

Projectile Launcher Steel projectile Meter Stick or Tape Measure Table clamp Carbon paper ExcelTM Paper Target

Activity 1: Determining Launch Velocity (Tabletop to tabletop launch)

1. Set up a projectile launcher at an arbitrary angle, other than 0 or 90o. The launcher should be adjusted so that it projects the projectile onto the tabletop. The angle that you set, , is the angle that you will use throughout the experiment. DO NOT point your launcher in the direction of the computer monitors!

2. Carefully measure the height from the tabletop to the launching position of the projectile. The manufacturer has placed a mark on the side of the launcher for the purpose of this measurement. This is the initial vertical position, y(0).

3. The launcher has three ranges: each range is determined by a click in the spring launcher and is also marked on the side of the launcher. Be sure to use the second click (medium range setting).

4. Fire the launcher and have a lab partner note the approximate position that the projectile strikes the table. Tape a piece of paper to the tabletop and place a sheet of carbon paper (carbon side down) on top of the taped paper. It is not necessary to tape the carbon paper to the table.

5. Fire for effect! Fire the launcher several times to obtain an average landing position. Estimate the center position of your pattern and measure the horizontal distance from this point to a point directly below the launching position of the projectile. This is the horizontal range, x.

6. Use the values for ay, x(t), x(0), , y(x) and y(0) to calculate the speed, v(0), at which the projectile left the launcher. (Assume that x(0)=0, and ay= 9.8 m/s2.)

7. Record the values of and the calculated launch speed, v(0), for use in the next activity as well as for the Post Lab.

Lab#3 ? 2D Kinematics

Activity 2: Determining Projectile Range (Tabletop to floor launch)

In the previous activity, you determined the speed and direction, hence the velocity, of a projectile being fired from your launcher. You will now use this information to predict the landing point of a projectile that is launched from the tabletop to the floor.

1. Turn your launcher so that it faces away from the table and towards an open space on the floor. Measure the vertical distance from the launching point of the projectile (on the side of the launcher) to the floor. This will be your new y(0). Record this value of y(0) for use in the Post Lab.

2. Use the value of v(0), y(0), and to determine the horizontal distance at which a target must be placed in order to be hit with your projectile.

3. Contact the TA when you are ready to fire your projectile.

IMPORTANT: Do not take any shots before the TA is contacted or you will have your launcher angle changed and you will need to begin again.

4. When the TA is ready, fire for effect. Your score will be the total points scored in five shots. You will be allowed one adjustment to the horizontal direction of the launcher.

Activity 3: Range vs. Angle

Theory predicts that, when a projectile starts and lands from the same vertical height, the maximum horizontal range should occur at 45o. In addition, there should be two distinct angles (complementary angles) of launch that would send a projectile to a particular range less than the maximum range.

1. Fire your projectile launcher (so that the projectile again lands on the tabletop) at different angles from 0? 90? at either 5? or 10? increments (depending on the amount of time available ? but include a data point for 45?). Record data for range and launch angle in a table, such as:

Launch Angle (degrees)

Range (meters)

2. Using ExcelTM , make a graph of Range vs. Angle.

3. Based upon your graph, does the maximum range occur at 45?? If not, where does it occur? 4. From your graph, generate an estimated list of at least 5 pairs of angle measures that yield the same range

values? Is each pair a set of complementary angles? Explain.

5. Make a record of your graph and/or data for use in the Post Lab.

Lab#3 ? 2D Kinematics

Name _____________________________________________ Section # ________________________________ Name _____________________________________________ Section # ________________________________ Name _____________________________________________ Section # ________________________________

Lab#3 ? 2D Kinematics

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