Physics HW 1

115L Lab One

Using Physical Principles and Measurements to Make a Prediction:

Target Practice with the Ballistic Pendulum

1 Introduction

Physics is an important science largely because it allows us to make accurate predictions of objects’ behaviors in different situations. This idea has been applied in the engineering of buildings, vehicles, and energy production. It is used to design aircraft, plan space missions and execute battle plans in warfare.

In the first part of this lab you will make use of two of the most valuable principles of physics, along with a couple basic measurements, to determine the speed of the ball launched by the spring gun in your ballistic pendulum. In the second part, you will calculate the point where the ball will strike when it is fired without the catching pendulum in place. You will fire the gun to test the accuracy of your predictions. Finally, you will do some simple analysis of the cause of any inaccuracy in your calculated targeting.

Tips for success:

• Make all your measurements as carefully as possible. The more accurate your measurements are, the closer you will come to hitting your target.

• Pay attention to units. Calculations require that all units match for the numbers to come out right. For example, a distance may be recorded in meters, centimeters, inches, miles, etc. The distance is fixed, but the value of the number used to record it can vary greatly.

2 Using Conservation of Momentum and Con- servation of Energy to measure the initial ve- locity of the ball

Test fire your ballistic pendulum 3 or 4 times, observing the parts and how the mechanism works. Be very careful not to get in the path of the ball!

Directly measuring the ball’s velocity as it is fired by the ballistic pendulum would be very difficult. However, since there are some physical properties that are conserved, meaning that the total amount cannot change–only the form can change or there can be a transfer from one object to another, the ball’s speed can be determined quite accurately with only a couple simple measurements

1

and a couple short calculations.

How could you go about measuring the ball’s speed as it launches from the ballistic pendulum?

Why would it be hard to use that method with this equipment?

2.1 Tracking the Energy

The act of firing the ball from the ballistic pendulum, and the different forms that the energy involved takes during the process, can be viewed in four distinct steps:

• Loading

• Firing

• Collision between the ball and pendulum

• Swing of the pendulum

Loading. To load the ballistic pendulum, you have to push the ball back against a very stiff spring. In the process of doing this you use some of the energy stored in your body. Once the pendulum is loaded, the energy that you gave up is now stored in the compressed spring.It would be difficult to make a measurement of the energy leaving your body during the compression of the spring, but it is quite easy to put a number on the energy once it is stored in the spring. The energy depends on how stiff the spring is, and how much it is compressed. This can be written as a mathematical expression:

Energy stored in a compressed spring: 1

2 kx2 (1)

In this expression, k is called the spring constant. It quantifies a property of the spring–how “stiff” it is.The term x is the measure of how much the spring is compressed.

Measure the distance that the spring is compressed when you load your ballistic pendulum. Make sure you measure from a consistent reference point. Record the measurement here so that you can use it later to determine k for your spring.

How precise is your measurement? In other words, how much could your be off the actual distance–more or less than?

What creates this uncertainty?

So during the loading, the energy involved moves from storage within the cells of your body to storage in the compressed spring. Because of this state of being “stored,” both of these forms of energy are known as potential energy.

As mentioned above, energy is a conserved quantity. The total amount of energy will always remain the same. It cannot be created or destroyed. It can be changed from one form to another. In practice, some energy will always be “wasted” during a transfer from one form to another. It isn’t really lost, it just ends up in some form, or forms, other than the intended one. Some transfer processes are more efficient than others.

The energy transfer during the loading of the ballistic pendulum is from a biologically stored potential energy to a mechanically stored potential energy. Where do you think some energy might escape to during the transfer?

(Think of the sweaty, grunting guy at the gym–maybe you are the sweaty grunting guy at the gym. Obviously a lot of energy goes into lifting several hundred pounds from one spot to a higher spot (another form of potential energy), but not all of it. Where does it end up?)

Firing. During firing, the energy is transferred from the potential energy of the compressed spring to the energy of the ball in motion. The energy associated with motion is called kinetic energy. The amount of energy depends on the size and speed of the moving object. Like the potential energy in the spring, the kinetic energy can be written mathematically:

Energy of an object in motion: 1

2 mv2 (2)

Here m is the mass of the moving object and v is its speed (v for velocity). Notice that the mass factor is in the expression in the first power (called a linear relation) and the speed factor is squared (quadratic).

If I’m lucky, I can throw a baseball about 45mph. A lot of major league pitchers can throw at 90mph. How many times more kinetic energy does a major league pitcher’s ball carry than mine?

The transfer of energy from compressed spring to moving ball is actually quite efficient, and now the principle of conservation of energy can be put to use.

Fire the ballistic pendulum a couple times. Pay close attention to the first part of the process, just as the ball flies off the spring.

Since the potential energy and kinetic energy both have mathematical expres- sions describing them, and we’re looking at a process where potential energy is

completely converted to kinetic energy, there is a way to write the principle of energy conservation itself as a mathematical expression. This is done by setting the two expressions (1 and 2) equal to each other:

Energy stored in the spring = Energy of the ball in motion

1

2 kx2 =

1

2 mv2 (3)

Since x and m can both be measured, this equation would allow the calculation of either k or v, if the other was known. But neither is,yet. That’s why you’ll need this next part...

Swing of the pendulum. The ballistic pendulum is set up so that as soon as the ball leaves the spring it embeds in the pendulum, which swings upward as it catches the ball. Partway up the arc there are some steps; and the pendulum has a ratchet lever so that it can swing out, but not back. This means that the pendulum will stop at the highest point in its swing. This provides the opportu- nity to make use of another form of potential energy, this one easily measured. Since gravity provides a constant force straight down to objects everywhere on, and slightly above, the surface of the Earth, any object experiencing that force has a potential energy stored. The higher up the object, the more energy it has available. You probably wouldn’t hesitate to drop your backpack with your laptop in it beside your chair as you sit down in class; but chances are you wouldn’t be willing to drop it off the third-floor landing just outside the lab. And, just like with kinetic energy, the amount of gravitational potential energy also depends on the size of the object involved. An acorn dropping from an oak tree onto a parked car will just bounce off and roll away. But if the limb that held the acorn drops–from the same height–hope they have good insurance. The mathematical expression is:

Gravitational potential energy: mgh (4)

The m stands for the mass again, and h is the height (above some specific point–you have to decide what point. It could be the floor in the lab, it could be the ground outside, it could be the table top. Gravitational potential energy is always relative to some other point). The g is the acceleration provided by the force of Earth’s gravity on any object. The acceleration is the same for any object. It actually depends on the distance from the center of the Earth, but near the surface of the Earth–the context of things in the lab, and at the height of people, buildings, trees, even airplanes and mountains–it is okay to consider it constant.

Now, by comparing the gravitational potential energy at the top of the arc to that at the bottom, you can say that the potential energy is zero at the bot- tom. Since nothing is moving once the pendulum sticks at the top of the arc, the kinetic energy there is zero. So the kinetic energy of the moving pendulum at the bottom of the arc is converted to gravitational potential energy at the top of the arc. This can be written as an equation:

Energy of motion at the bottom = Potential energy at the top

1

2 mv2 = mgh (5)

Fire the pendulum again. Pay close attention to the pendulum’s motion from stationary at the bottom to locked at the top of its arc.

The energy involved in this process follows the path from potential energy in

your body, to potential energy in the compressed spring, to kinetic energy of the moving ball, to kinetic energy of the moving pendulum, and finally gravitational potential energy of the pendulum at the highest point of its arc.

The left side of equation 5 looks identical to the right side of equation 3. If they were describing the same thing, you’d have the v that you need, by solving equation 5. Why aren’t the two vs the same?

What else is different between the two kinetic energies?

What happens between the situation described in equation 3 and that described in equation 5 that hasn’t been considered yet?

The collision.Take a look at figures 1, 2, and 3. They show the three stages of the ball’s trip as the ballistic pendulum is fired. You’ve looked at using conservation of energy to write equation describing how potential energy stored in a spring is completely turned into the kinetic energy of a moving ball, as shown in figure 1. And you’ve looked at the equation describing the process shown in figure 3, where the kinetic energy of the swinging pendulum becomes gravitational potential energy as the pendulum ends up higher than where it started. However, the situation in figure 2 is a little different. This is the exact moment where the ball contacts the pendulum and becomes embedded. Before the collision the energy is in the form of the kinetic energy of the moving ball; after the collision the energy is still kinetic, now the moving combination of ball and pendulum. But the assumption that the kinetic energy following the collision is equal to the kinetic energy just before the collision is not true here.

This type of collision, called “inelastic” because the two objects stick together rather than bouncing off each other, allows some of the initial kinetic energy to be converted into other types of energy not related to the motion or position of the objects.

One name for the form that energy takes during an elastic collision is “internal energy”. Have you noticed anything different about the way the ball feels after a firing, or especially after a couple consecutive firings, that would indicate how internal energy might show itself?

Figure 1: Before the collision the ball carries kinetic energy that was transferred from the potential energy stored in the compressed spring. The pendulum is stationary, and at its lowest point.

Figure 2: At the moment of the collision the ball’s kinetic energy becomes kinetic energy of the combined ball and pendulum, but some of the energy is transferred to other forms. This is the step where conservation of energy cannot be used as a tool in the analysis of the process.

Figure 3: Just after the collision the ball and pendulum are in motion with whatever kinetic energy they are left with. As the pendulum rises in its arc, it slows down. Kinetic energy is transferred to gravitational potential energy. At some point, all the initial kinetic energy has become potential energy and the pendulum comes to rest. Conservation of energy is a good approximation again after the collision.

2.2 Collect the Post-Collision Energy Data

You will need to make use of a different principle of physics to make the connec- tion between the energy of the ballistic pendulum before and after the collision, and make the determination of ball’s speed leaving the spring. But since con- servation of energy is a good assumption during the pendulum’s swing following the collision, the gravitational potential energy at the end of the swing will give you a way to get the speed of the combined ball and pendulum immediately after the collision. The gravitational potential energy is easily measured.

• Choose a reference point on that pendulum that will be easy to use for measurements in both the lowest and highest positions.

• Measure the height of the pendulum at its lowest point. Enter the value in the table below.

• Fire the pendulum.

• Measure the height at the high point. Enter the value in the table below.

• The change in height for the pendulum, the difference between the high point and low point, is h. Enter this value in the table.

• Repeat these steps for five total trials.

Trial Bottom Top h=(Top-Bottom) 1 2 3 4 5

Since the height is not the same for each time the pendulum is fired, the energy given to the pendulum by the ball must not be the same each time. You will use the average change in height for the pendulum to calculate an average kinetic energy, and thus velocity, for the pendulum at the start of its swing.

Calculate the average h here:

Using equation 5, 1 2 mv2 = mgh, and your average value for h, calculate

the velocity of the pendulum at the start of its swing. Show your work here:

2.3 Using Momentum to Find the Velocity Transferred in the Collision

The idea that will allow you to determine the velocity of the ball before the collision is conservation of momentum. Momentum, like kinetic energy, depends on an object’s size and speed. But momentum also includes the information about the direction the object is travelling in. Technically, direction information is the difference between velocity and speed. The speed of an object is how fast it is moving. The velocity of an object is how fast it is moving and in what direction. The mathematical expression for momentum is just mv, with m being the object’s mass and v the objects speed in a certain direction.

The really useful thing about momentum for calculations is that, when you’re considering motion in one certain direction, the total amount of momentum in that direction will always be the same. This applies even if more than one object is involved, as long as you remember to look at one specific direction. The principle will hold, even if there is an inelastic collision—as with the ball and pendulum in figure 2. The conservation principle means you can write an equation describing the collision: total momentum before = total momentum after. Using figures 1 and 2 as references for before and after, respectively, the conservation of momentum equation for the collision is:

Momentum before collision = Momentum after collision

mv1 = (m + M)v2 (6)

The left side has only the ball’s mass and velocity, since the velocity of the pen- dulum before the collision is zero. The right side has the combined mass of the ball and pendulum, now stuck together as one object, and the new velocity of that combination. Remember, the first motion of the pendulum is in the exact same direction that the ball was moving before the collision. After it starts moving the pendulum follows an arc, but that is because it is attached to a pivot and can’t continue in a straight line. The motion of the pendulum along the arc makes using momentum impractical for that part of the process, which is why you used energy conservation to analyse that portion. But conservation of momentum is just what you need to treat the exact moment of the collision.

You will need the masses of the ball and pendulum. Use a triple beam balance to get the ball’s mass. The pendulum’s mass is marked on the apparatus.

Use equation 6, mv1 = (m + M)v2, to find the velocity of the ball before the collision, v1. Remember, v2, the velocity of the ball after the collision, is the velocity that you calculated using conservation of energy.

You also have the information now to find the spring constant, k, using conservation of energy. Use equation 3, 1

2 kx2 = 1

2 mv2, to find k for your spring.

3 Using Physics and Calculation to Determine the Ballistic Pendulum’s Target Range

Now that you’ve determined the ball’s initial velocity as it is fired from the bal- listic pendulum, you’re ready to take a look at a couple more physical principles that will allow a calculation to predict the point where the ball will land when fired without the catching pendulum in place. Then you’ll test your prediction by placing a target and seeing how close you come to hitting the bullseye.

First you will need to make use of an idea very much related to what was discussed during the discussion of momentum. Remember, momentum carries

information about a specific direction. When a ball is moving it has momentum, mv, in the direction is moving. In the direction perpendicular to its motion, its momentum is zero–because it has zero velocity in that direction. Force is a physical property that, like momentum, includes a direction. Force is actually very closely related to momentum. It is the change of momentum.

Drop your ball from about the same height as the table. What was its speed right before you dropped it?

What does that mean its momentum was?

How about right as it hit the floor? Speed?

Momentum?

Now, this may seem obvious, but its important–what direction was it moving right as it hit the floor?

Remember, momentum includes direction. Was there a change in the ball’s vertical momentum after you dropped it?

Was there a change in the ball’s horizontal momentum?

The only force acting on the ball once you let go of it was due to gravity. Gravity is acting straight down, the ball’s momentum in the vertical direction changed. It sped up until it hit the floor. But nothing made its horizontal momentum change. It dropped straight down. The force only acts in one direction—the momentum only changed in one direction.

Drop the ball from table height again. Make an estimate of how long it takes it to hit the floor.

If the ball drops from table height to the floor this is how long it will take it to fall. This is true even if it has some motion in the horizontal direction at the start. The force due to gravity won’t cause any change in the horizontal momentum because it is a change of the vertical momentum–perpendicular to the initial horizontal motion.

Roll the ball off the edge of the table. Estimate the time it takes it to hit the floor. Does it take the same amount of time as when you just dropped it?

Repeat a few times with different speeds of rolling. Can you tell any difference in the time it takes the ball to hit the floor?

Does the speed of rolling change where the ball lands?

When you want to predict the location of an object at a certain time, there is an equation that is very useful. Again, the equation applies to motion in one specific direction. Since you’re interested in motion in two directions—vertical and horizontal—you’ll need to use the equation twice. The equations are func- tions of time. In other words, if you plug in a specific time, the equations will give you the location of the object at that time. It is common to use x for the horizontal direction, and y for the vertical direction. Here are the equations:

Horizontal position:

x = x0 + v0xt + 1

2 axt

2 (7)

Vertical position:

y = y0 + v0yt + 1

2 ayt

2 (8)

Since you want to place your target where the ball will hit, and you know the vertical position where that will happen—the floor—take a closer look at the horizontal, x, equation first.

The general form for the equation for the horizontal position is equation 7, x = x0 + v0xt +

1 2 axt

2.

• x on the left side of the equation is what you want to know. Its your dependent variable.

• On the right side of the equation, x0 is the initial position of the ball. You’re interested in how far it goes from where it is now. What’s a good choice for the value of x0?

• v0x is the ball’s initial velocity in the x direction. In what direction is the ball launched by the ballistic pendulum’s spring?

• ax is the acceleration of the ball in the x direction. Acceleration is the change of velocity in a specific direction. It is very closely related to force being the change of momentum in a certain direction. What is your acceleration in the x direction?

• t is the time. It is your independent variable.

So, by calling your starting horizontal position zero, and realizing that there is no force (and thus no acceleration) in the horizontal direction, the equation for

the horizontal position of the ball becomes:

x = v0xt (9)

With v0x being your measured value of the ball’s speed as it is launched by the spring.

How do you know what value to use for t?

You are interested in the moment the ball hits the floor, so the length of time from the release of the ball until the moment it hits the floor is the t that you need. Since the direction of motion from the table top to the floor is vertical, the information should come from the y equation. Take a closer look at the y equation, 8.

The general form for the equation for the vertical position is equation 8, y = y0 + v0yt +

1 2 ayt

2.

• y on the left is the final position in the vertical direction. What will it be?

• On the right side, y0 is the initial vertical position of the ball. You’ll need to measure this.

• v0y is the ball’s initial velocity in the vertical direction. Does the spring launch give the ball any speed up or down?

• ay is the vertical acceleration. This is related to the force from gravity. It is the same g that appeared in the expression for the gravitational potential energy. This equation takes direction into consideration–assuming you measured up from the floor to the ball for y0, what sign should you assign to g to account for the direction it acts in?

• t is the same t as in the x equation. If you know the rest of parts of the y equation, you can get t and solve the x equation.

If you haven’t done it already, measure the height of your ball above the floor, when it is loaded in the ballistic pendulum. This is your y0.

Using your measurement for y0, solve the y equation for t: 0 = y0 − 12gt

2

Now that you know when the ball will hit the floor, you can use the x equation, equation 9, to find out where it will hit.

Use your t and v0x in the x equation to find where to place the target. x = v0xt

Carefully measure from the ball’s initial position out horizontally the distance x that you have calculated, and place your target.

Fire!

• Mark the spot where the ball actually hit, and measure the distance it is away from your target’s bullseye.

• Shoot the ball a total of 5 times and get the distance that you miss the bullseye for each.

• Get the average value for 5 “misses”.

One way to rate the accuracy of a measurement is to compare the amount the measurement is off from the actual value to the value of the measurement. You can do this by calculating what percent the distance of your “miss” is of the total distance you predicted.

Divide your average “miss” distance by the x you had calculated, then multiply by 100.

This is the percent of your prediction that you were off by.

An important part of using physics calculations to make predictions is choosing which factors must be considered, and which can be ignored. The more details that are considered, the more accurate the prediction will be; but details also increase the complication of the calculation. Often, something that has a small effect physically can be left out of consideration in the calculation and the result will still be accurate enough for the purposes. Physicists always make choices about what information to include in their calculations.

Were your “misses” long or short?

What factors that were ignored in setting up the calculations for your prediction might account for the trend in your misses?

Can you think of a quick adjustment you can make to the apparatus that will get you closer to the target?

Try it.