MOTION IN 2 DIMENSIONS

For motion in 2 dimensions we can calculate components of velocity and acceleration by differentiating x and y as functions of time.

We can then use Pythagoras to get the total value

PROJECTILE MOTION

Ball Projectile Motion : Some Observations

So far we have been dealing with horizontal motion along a straight line and vertical motion along a straight line separately. In this unit and the next we would like to study motions in a plane such as the motion of a cannon ball or the circular orbit of the planet Venus.

If a cannon ball is shot off a cliff with a certain initial velocity in the x-direction and y-direction, it will continue to move forward in the x- direction and at the same time fall in the y-direction as a result of the attraction between the earth and the ball. The two-dimensional motion that results is known as projectile motion.

Why is it that the projectile keeps the same horizontal velocity?

Interactive Physics Coconut Kick

This simulation investigates the use of vectors to solve physics problems.

Situation A

Kenji and Felecia find some coconuts near the edge of a cliff. They know they can calculate the height of the cliff by finding the time it takes the coconut to fall to the ground and then using the formula d = 1/2gt^2. They wonder if the formula will work for a coconut kicked off the cliff instead of dropped straight down over the edge. Kenji and Felecia design an experiment to find the answer. Kenji drops one coconut over the edge at the exact moment Felecia kicks the other coconut off the cliff.

Make a Prediction

1. Predict which coconut will hit the ground first - the coconut dropped over the edge or the coconut kicked outward off the cliff. Assume there is no air resistance.

• The dropped coconut will hit the ground first.
• The kicked coconut will hit the ground first.
• Both coconuts will hit the ground at the same time.

Simulation: Set Initial y velocity to 0.00 m/s. Set Initial x velocity to 5.00 m/s. Set all Vector Displays to Off. Click the Tracking On button. Click the Run button and observe the simulation.

2. List the forces acting on each coconut as it falls. In which direction, x or y, do these forces act?

3. Was your prediction correct? Explain the results in terms of the force vectors acting on each coconut.

4. How do the vertical positions of the falling coconuts compare?

5. Using the results of the simulation and the formula d = 1/2agt^2 , calculate the height of the cliff (ag = 10.00 m/s^2).

Situation B

Kenji and Felecia decide to investigate what will happen when one coconut is thrown vertically upward while the other is kicked upward and outward. Assume both the coconuts have the same initial vertical velocity.

Make a Prediction

6. Predict which coconut will hit the ground first - the coconut thrown upward or the coconut kicked upward and outward. Assume there is no air resistance. Place a check next to your prediction.

• The coconut thrown upward will hit the ground first.
• The coconut kicked upward and outward will hit the ground first.
• Both coconuts will hit the ground at the same time.

Simulation: Click the Reset button. Set Initial y velocity to 6.00 m/s. Set Initial x velocity to 8.00 m/s. Set the Force of gravity and Velocity Vector Displays to On. Click the Tracking On button. Click the Run button and observe the simulation. Note: When the Vector Displays are on, two additional coconut images will appear on the screen. These images are used to show the Force of gravity vector, FG, and the Velocity vector, V, associated with each of the coconuts as they fall.

7. Was your prediction correct? Explain the results in terms of the force vectors acting on each coconut.

8. Explain what happens to the Force of gravity vector as the coconuts fall.

9. Describe what happens to the Velocity vector for each of the coconuts as they fall. Explain why the Velocity vector for the kicked coconut changes magnitude and direction during the fall.

There are many other Interactive Physics Simulations you can play with to simulate projectiles. See your instructor if you want to use them.

Artillery Range: - A Computer Simulation

In this simulation you will work with the equations of projectile motion to try to hit a target position. Run the software Projectile . There are a number of "Challenges" for you to solve. For each challenge that does not have air resistance, practice a little then make calculations so that you can make a direct hit. Keep it up till you are successful. For those that involve air resistance, comment on its effect. Be quantitative if possible.

You will need the following equations for the x and y coordinates of a projectile and the x and y components of the velocity. Your calculator for solving quadratics will be useful. These equations are valid for constant acceleration only:

x = xo + vo cos q t (notice there is constant velocity - no acceleration)

y = yo + vo sin q t - 1/2agt^2 (notice there is constant acceleration)

vx = vo cos q

vy = vo sin q - agt

DO CHALLENGES 1 - 4

(do more if you like; most students try #s 8 and 9)

Show calculations clearly for each one that doesnt have air resistance.

Say whether you were successful on the first attempt.

CHALLENGE 1 CLIFF NO AIR RESISTANCE

Horizontal Launch Calculations (set the angle = 0)

Angle Launch Calculations (pick an angle other than zero)

CHALLENGE 2 CLIFF WITH AIR RESISTANCE

CHALLENGE 3 GROUND NO AIR RESISTANCE

Angle Launch Calculations

CHALLENGE 4 GROUND WITH AIR RESISTANCE

CHALLENGE 5 GROUND WITH AIR RESISTANCE & WIND

CHALLENGE 6 RANGE vs. SPEED

Use one angle between 20° and 60°, take enough data to sketch a graph of range vs. velocity squared.

CHALLENGE 7 RANGE vs. ANGLE

Keep the speed constant and vary the angle. Is the angle for maximum range what you would expect:

CHALLENGE 8 CLIFF, MOVING TARGET, Horizontal Launch

Hint: The equation for the motion of the target is x = xo + vt ; the projectile is x = xo + vo cos q t, when the projectile hits that target the x-coordinates and the y-coordinates are equal.

CHALLENGE 9 GROUND, MOVING TARGET

 

GI JOE PROJECTILE METHOD (Formal Lab Report)

OBJECTIVE

To determine the range of a projectile as a function of projection angle.

MATERIALS

Projectile gun with bullet, paper, carbon paper, meter stick or tape measure.

PROCEDURE

STEP 1 Finding the speed of the ball

1. You probably should do this experiment outdoors(if the weather is cooperative) so that the projectile doesn't hit anything; otherwise an empty hallway may be adequate. Set the gun apparatus on a table in such a way that it can be fired at an angle 0^o relative to the horizontal. Aim the projectile where it won't hit someone or something. At some angles it will have a range of around 5 meters and a maximum height of nearly 3 meters.

y = yo + vo sin q t - 1/2agt^2 , x = xo + vo cos q t

use yo = height then when it hits the ground y = 0

use xo = 0

• Measure the height of the bullet from the floor. Shoot the ball at 0^o several times; use the range, X- position when it hits the ground, to find the initial speed of the bullet.

STEP2 Projected at Same Height as Lands: Measuring Range

• After you have found the initial velocity of the projectile using the table and zero degrees, you will shoot the projectile from ground level and use a box or a few books so that it lands at the same altitude it was shot; yo = yf = 0. Place carbon paper on the books or box and put the box at the expected impact point, then run two or three trials for each angle in a chart like the one below. Do this for each of the angles (but not zero degrees). Station a lab partner near where the projectile strikes the box. He is to measure the average range of the two or three trials. Measure the average range for each angle of projection and record the data.

STEP3 Projected at Same Height as Lands: Calculating Range

• If you have taken trigonometry, using Trig identities you should be able to show that the range is

. Use the initial velocity, vo, from procedure 2 above at zero degrees in all your calculations and fill in the Theoretical Displacement in your chart.

• On one graph plot both the theoretical and the actual range versus the angle of projection, drawing a smooth curve through each series of points.

STEP 4 Projected from a Cliff: Measuring Range

• For the chart below you will shoot the projectile from a ledge or table that is at least 80 cm high. Run two or three trials for each angle and record the actual height and range in the chart. [Keep height same throughout]

• It's too hard to calculate the range in this case, so what we are going to do is to make a table of height vs time for each angle, and see at what time the projectile hits the ground. Then find the x value at that time.

For the theoretical part of the chart, in Excel use the equations for projectile motion to predict where the bullet will land for the angles 20^o, 30^o, 40^o, 50^o, 60^o, and 70^o when given the initial height (see the example below). Type your measured height and initial velocity into the Excel sheet to obtain theoretical values. Enter a height in Cell A2. Enter times, 0, 0.1 By tenths from A3 down. Enter your angle in B2, convert it to radians and put the result in C2. Enter an initial velocity in cell D2. You will have found this initial velocity in procedure 2. Now enter the equations for y and x in columns below B3 and C3 with appropriate labels. Use $A$1 and $C$1 to refer to the height and angle. Use 9.81 m/s^2 for ag. Copy the formulas down the columns. When y = 0 the ball has hit the ground. The x coordinate when y = 0 is the range.

• For angles of 10 - 70 degrees use the spreadsheet to find the range.
• On one graph plot both the theoretical and the actual range versus the angle of projection, fitting a smooth curve (parabola) through each series of points.
• From the two graph, determine the angle of projection for the maximum range and record the below. Compare with the theoretical prediction of tan q =

RANGE:

From Actual Graph_________ From Theory Graph _________ From Formulas ______

STEP 6 Shoot Ken!

Consider completing the graphs before you do this part since it is much easier than doing the calculation. Have your partner place a target ( such as Ken, Barbie or GI Joe) on the ground at some position within the maximum range of the projectile. By measuring the horizontal distance to the target predict the angle(s) required to hit the target. Aim and shoot for both of your predicted angles. Record your accuracy.

Horizontal distance to target __________

Predicted Angles ____________

How far from the target did your closest projectile land? ___________; Which angle was closest? _________

Evaluate your performance as an artillery person. Why is the smaller angle usually more accurate than the larger one?

Monkey Hunter

• Observe the demonstration of the Monkey and the hunter.

The essential fact is that all the pictures

which science now draws of nature . . .

are mathematical pictures.

Sir James Jeans

Circular Motion And Centripetal Force

Moving in a Circle at a Constant Speed

When a race car speeds around a circular track, or when David twirled a stone at the end of a rope to clobber Goliath, or when a planet like Venus orbits the sun, they undergo uniform circular motion. Understanding the forces which govern orbital motion has been vital to astronomers in their quest to understand the laws of gravitation.

Uniform circular motion. A ball moving at a constant speed in a circle of radius r.

Let's begin our study with some very simple considerations. Suppose an astronaut goes into outer space, ties a ball to the end of a rope, and spins the ball so that it moves at a constant speed.

Uniform Circular Motion

(a) What is the speed of a ball that moves in a circle of radius r = 2.5 m if it takes 0.50 s to complete one revolution?
(b) The speed of the ball is constant! Would you say that this is accelerated motion?
(c) What is the definition of acceleration? (Remember that acceleration is a vector!)
(d) Are velocity and speed the same thing? Is the velocity of the ball constant? (Hint: Velocity is a vector quantity!)
(e) In light of your answers to (c) and (d), would you like to change your answer to part (b)? Explain.

By now you should have concluded that since the direction of the motion of the ball is constantly changing, its velocity is also changing and thus it is accelerating.

The acceleration associated with uniform circular motion is known as centripetal acceleration.

 

Using Vectors to Find the Direction of Centripetal Acceleration (optional)

(a) Determine the direction of motion of the ball shown in diageam in text if it is moving counter-clockwise at a constant speed. Note that the direction of the ball's velocity is always tangential to the circle as it moves around. Draw an arrow representing the direction and magnitude of the ball's velocity as it passes the dot just before it reaches point A. Label this vector .

(b) Next, use the same diagram to draw the arrow representing the velocity of the ball when it is at the dot just after it passes point A. Label this vector .

(c) Find the direction and magnitude of the change in velocity as follows. In the space below, make an exact copy of both vectors, placing the tails of the two vectors together. Next, draw the vector that must be added to vector to add up to vector ; label this vector . Be sure that vectors and have the same magnitude and direction in this drawing that they had in your drawing in part (a)!

(d) Now, draw an exact copy of on your sketch in part (a). Place the tail of this copy at point A. Again, make sure that your copy has the exact magnitude and direction as the original in part c).

(e) Now that you know the direction of the change in velocity, what is the direction of the centripetal acceleration, ?

(f) If you redid the analysis for point B at the opposite end of the circle, what do you think the direction of the centripetal acceleration,, would be now?

(g) As the ball moves on around the circle, what is the direction of its acceleration?

(h) Use Newton's second law in vector form () to describe the direction of the net force on the ball as it moves around the circle.

(i) If the ball is being twirled around on a string, what is the source of the net force needed to keep it moving in a circle?

How Does ac Depend on v and r?

(a) Do you expect you would need more centripetal acceleration or less centripetal acceleration to cause an object moving at a certain speed to rotate in a smaller circle?

(b) Do you expect you would need more centripetal acceleration or less centripetal acceleration to cause an object to rotate at a given radius r if the speed v is increased?

You should have guessed that it requires more acceleration to move an object of a certain speed in a circle of smaller radius and that it also takes more acceleration to move an object that has a higher speed in a circle of a given radius. Lets use the definition of acceleration in two-dimensions and some accepted mathematical relationships to show that the magnitude of centripetal acceleration should actually be given by the equation

[Eq. 1]

In order to do this derivation you will want to use the following definition for acceleration

[Eq. 2]

Derivation of Formula for Centripetal Acceleration (optional)

(a) Refer to the diagram in text. Explain why, at the two points shown on the circle, the angle between the displacement vectors at times t1 and t2 is the same as the angle between the velocity vectors at times t1 and t2. Hint: In circular motion, velocity vectors are always perpendicular to their displacement vectors.

(b) Since the angles are the same and since the magnitudes of the displacements never change (i.e. r= r1 = r2) and the magnitudes of the velocities never change (i.e. v = v1 = v2), use the geometry of triangles to show that

(c) Now use the equation in part (b) and the definition of < a > to show that

(d) The speed of the object as it rotates around the circle is given by

Is the change in arc length, D s, larger or smaller than the magnitude of the change in the position vector, D r? Explain why the arc length change and the change in the position vector are approximately the same when D t is very small (so that the angle q becomes very small) i.e. why is D s = D r?

(e) If D s = D r, then what is the equation for the speed in terms of D r and D t?

(f) Using the equation in part c), show that as D t > 0, the instantaneous value of the centripetal acceleration is given by the equation

(g) If the object has a mass m, what is the equation for the magnitude of the centripetal force needed to keep the object rotating in a circle (in terms of v, r, and m)? In what direction does this force point as the object rotates in its circular orbit?

 

Experimental Verification of the Centripetal Force Equation

The theoretical considerations in the last activity should have led you to the conclusion that, whenever you see an object of mass m moving in a circle of radius r at a constant speed v, it must at all times be experiencing a net centripetal force directed toward the center of the circle which has a magnitude of

[Eq.3]

Lets check this out. Does this equation really work for an external force?

To do this experiment you will need the following equipment:

• A smooth, level area of r about 2 m (use about 1.5 m for the radius)
• A cart with bearings to support 2D motion
• A post or anchor person in the center of the area
• A spring scale (15 kg or 150 N)
• 2 smooth ropes
• A stop watch
• A belt with rings on the sides for the rider
• A bathroom scale (for persons mass)

You should attach the spring scale between the rider and the rope. The end of the rope should be attached to the cart rider's belt or held by the rider. The second rope should be attached to the cart in a direction which is perpendicular to the original rope. See the diagram in text for details.

(a) If a puller applies a force on the cart which is always tangent to the circle and which is just sufficient to overcome friction in the cart and maintain the cart's motion at a constant speed, what is the net force in a direction perpendicular to the circle? Remember Newton's first law!!??

(b) You and members of your group should practice pulling each of your members around at a constant speed. The rider should hold on to the rope and close his or her eyes and concentrate on feeling the centripetal force.

(c) When you hold on to the center rope while riding on the cart, what is the direction of the net force you feel on you (include consideration of the forces in the radial direction and in the tangential direction.) Does the force you feel seem to increase as you rotate faster? As you are rotate in a smaller circle? Explain.

(d) Next, keep the radius (about 1.5 - 2.0 m) to the center of the cart and the mass of the cart and rider constant and take data for several different orbital speeds with the rope attached to the belt the rider is wearing. Set up a spread sheet and compare the theoretical value of Fc calculated using equation 3 with the experimental value obtained by reading the average force in newtons recorded on the spring scale. Explain your procedures, show sample calculations, and present your results.

Note: When reading the force scale you should record the amount of wobble around an average value and use it as an estimate of the most significant uncertainty in your measurements.

(f) Within the limit of experimental uncertainty, how well does your experimental data support the hypothesis that Fc is a function of v^2?

(g) Lets look at agreement between experiment and theory another way. How well does each measured force agree with the corresponding calculated force shown in your data table?

(h) Discuss the major sources of uncertainty in this experiment. There are plenty!

 

REPEATING THE EXPERIMENT A LITTLE MORE ACCURATELY:THE RUBBER STOPPER

PURPOSE

(1) To study the nature of centripetal force.

(2) To experimentally demonstrate the relationship between centripetal force, mass and velocity.

MATERIALS

Glass tube 15 cm long; masking tape; nylon fishline; rubber stoppers; masses; paper clip or tape ; clock or stopwatch; triple beam balance.

INTRODUCTION

An object moving with changing speed in the same direction is undergoing acceleration. If an object moves with constant speed but in changing directions, it is also undergoing acceleration. Both types of acceleration require forces. A change in direction is called centripetal acceleration, and the force producing it is called centripetal force. The equation relating centripetal force, mass, and velocity is Fc = mv^2 /r where Fc is the centripetal force, m is the mass of the moving object, v is its speed, and r is the radius of the orbit of the object. In this experiment, each of the factors in this equation will be varied as an object is whirled on the end of a string. Centripetal force will be supplied by masses tied to a string that passes through a vertical tube. The effect of gravity on the whirling object is offset by the resulting angle of the string with the horizontal. Thus r can be taken as the length of the string between the tube and the center of the object (even though the string is not perpendicular to the tube) without introducing a significant error. Of course, it is possible for an object to have accelerations of speed and direction at the same time. This is the case with the planets, which move around the sun in elliptic orbits.

PROCEDURE

Each group will perform one part of the following procedure. The data will then be shared.

1. Attach a #6 rubber stopper of mass 20 grams (as close as you can make it, by cutting off a little or adding a little tape) securely to one end of a fishline; feed the fishline through the glass tube. Fasten 100 g to the other end, hold it in one hand and hold the glass tube in the other. Fasten a paper clip or tape to the fishline so the radius of the circle will be 60 cm from the top of the tube to the center of the stopper. Whirl the rubber stopper in a horizontal circle by revolving the tube. Slowly release the 100 g mass and adjust the speed of revolution so that the paper clip or tape marker stays just (1-2 mm) below the bottom of the tube. Make several trial runs before recording any data. When you have learned how to keep the horizontal rotational speed and position of the marker constant, have a partner measure the time required for 20 or 30 revolutions. Calculate the time of a single revolution and record as the period. Also record the centripetal force (weight of the 100 g).

Repeat this procedure, but increase the centripetal force by adding about 50 g (two big washers). Record the resulting data. Run four additional trials using still larger centripetal forces in 50 g increments.

2. Run five of trials in which the radius (60 cm) and force (0.980 N) are kept constant, but the mass of the whirling object changes. Use Stoppers from #4 - #8.

3. Run a series of trials in which the mass (use a #6 stopper with a mass 20 grams (as close as you can make it, by cutting off a little or adding a little tape)) of the object and the force (0.980 N) are kept constant, but the radius changes ( use 40 cm, 50 cm, 60 cm, 70 cm, 80 cm).

Record the appropriate data.

!DATA / RESULTS

1. Calculate the velocity from the circumference and period.

2. Using the centripetal force equation, calculate the mass times the acceleration for each trial. Compare these values with the values of force and explain any discrepancies

3. GRAPHS ( do those assigned to you; consider obtaining results from other groups who used the same controls as you did and combine on one graph.)

Estimate the uncertainty in your measurements and write at the top of the data chart. Find the equation of your graph. Interpret the intercept of your graphs.If it is not zero, should it be zero? Analyze the slope and compare it to what you would expect?

4.Based on the graphs, what is the relationship between the centripetal force, mass, radius and the speed of a body in circular motion? Do your graphs support the textbook equation or do they show it to be wrong?

5. If the tape or marker touches the tube, how does this affect results?

6. If there is a lot of friction between the tube and the string, how does this affect results?

Newton's Apple

Newton used the idea of centripetal acceleration and the observation that the earth moves in an approximately circular orbit around the sun to conclude that the earth's orbit is a result of a gravitational force between the sun and the earth.

His biographer, Richard Westfall, commented that even if Newton had died in 1684, before completing the Principia, the world would have recognized him as a genius. However, many intellectual historians consider Newton's most profound contribution to science to be his recognition that the laws that govern the orbital motion of planets are the same laws that govern the falling of objects near the earth and the linear acceleration of objects that are pushed or pulled with constant forces.

The image of Newton having this idea as a sudden insight while sitting under an apple tree is probably mythical. However, it leads to some interesting modern facts. Nabisco now distributes both Fig and Apple Newtons. Near the surface of the earth the force on an average sized apple is about one Newton. Think of this the next time you eat Apple Newtons!