introduction to mechanics relative motion and...

Introduction to MechanicsRelative Motion and Projectiles

Forces

Lana Sheridan

De Anza College

Feb 20, 2020

Last time

• trajectory equation

• another projectile example

Overview

• relative motion and projectiles

• forces

• net force

Relative Motion And Projectile Motion

Observer on the skateboard sees the ball fall straight down.

EXAMPLE 4–3 Dropping a BallA person skateboarding with a constant speed of 1.30 m/s releases a ball from a height of 1.25 m above the ground. Giventhat and find x and y for (a) and (b) (c) Find the velocity, speed, and direc-tion of motion of the ball at

Picture the ProblemThe ball starts at and Its initial ve-locity is horizontal, therefore and

In addition, it accelerates with the acceleration dueto gravity in the negative y direction, and moveswith constant speed in the x direction,

StrategyThe x and y positions are given by and respectively. We simply substitute time into these expressions.Similarly, the velocity components are and vy = -gt.vx = v0

y = h - 12 gt2,x = v0t

ax = 0.ay = -g,

v0y = 0.v0x = v0 = 1.30 m/s

y0 = h = 1.25 m.x0 = 0

t = 0.500 s.t = 0.500 s.t = 0.250 sy0 = h = 1.25 m,x0 = 0

4–3 ZERO LAUNCH ANGLE 85

PROBLEM-SOLVING NOTE

Identify Initial Conditions

The launch point of a projectile deter-mines and The initial velocity of aprojectile determines and v0y.v0x

y0.x0

y

h h

v0

OOxx

y

! > 0 v0! = 0

(a) (b)

O x (m)

y (m)10 9 8 7 6 5 4 3 2 1

21 643 5 7▲ FIGURE 4–5 Trajectory of a projectilelaunched horizontallyIn this plot, the projectile was launchedfrom a height of 9.5 m with an initialspeed of 5.0 m/s. The positions shown inthe plot correspond to the times

Note theuniform motion in the x direction, andthe accelerated motion in the y direction.

t = 0.20 s, 0.40 s, 0.60 s, Á .

FIGURE 4–4 Launch angle of aprojectile(a) A projectile launched at an angleabove the horizontal, A launchbelow the horizontal would correspondto (b) A projectile launchedhorizontally, In this section weconsider The next section dealswith u Z 0.

u = 0.u = 0.

u 6 0.

u 7 0.

▲

x

y

O

h = 1.25 m

v0

g

ball is given by

and

This is illustrated in Figure 4–3.The initial velocity is horizontal, corresponding to in Figure 4–4. As a

result, the x component of the initial velocity is simply the initial speed,

and the y component of the initial velocity is zero,

Substituting these specific values into our fundamental equations for pro-jectile motion (Equations 4–6) gives the following simplified results for zerolaunch angle

4–7

Note that the x component of velocity remains the same for all time and thatthe y component steadily decreases with time. As a result, x increases linearlywith time, and y decreases with a dependence. Snapshots of this motion atequal time intervals are shown in Figure 4–5.

t2

y = h - 12 gt2 vy = -gt vy

2 = -2g¢y

x = v0t vx = v0 = constant vx

2 = v0

2 = constant

1u = 02:v0y = v0 sin 0° = 0

v0x = v0 cos 0° = v0

u = 0

y0 = h

x0 = 0

continued on next page

WALKMC04_0131536311.QXD 11/16/05 17:57 Page 85

Another observer on the sidewalk sees the ball as a horizontallylaunched projectile.


#73, page 108

To decide who pays for lunch, a passenger on a moving train tossesa coin straight upward with an initial speed of 4.38 m/s andcatches it again when it returns to its initial level. From the pointof view of the passenger, then, the coin’s initial velocity is(4.38 m/s) y. The train’s velocity relative to the ground is(12.1 m/s) x.

(a) What is the minimum speed of the coin relative to the groundduring its flight? At what point in the coin’s flight does thisminimum speed occur? Explain.


#73, page 108


(b) Find the initial speed and direction of the coin as seen by anobserver on the ground.


#73, page 108


(c) Use the expression for h = ymax to calculate the maximumheight of the coin, as seen by an observer on the ground.


#73, page 108


(d) Calculate the maximum height of the coin from the point ofview of the passenger, who sees only one-dimensional motion.

Forces

Up until now we have predicted the motion of objects fromknowledge of their motional quantities, eg. their initial velocities,accelerations, etc.

We did not consider what the causes of this motion might be. Wenow will think about that.

We will understand forces as the cause of changes in the motionof objects.

Forces are a “push” or “pull” that an object experiencesbecause of an interaction.

Forces are vectors.

Forces

Two types of forces

• contact forcesanother object came into contact with the object

• field forcesa kind of interaction between objects without them touchingeach other

Forces

Force type examples:112 Chapter 5 The Laws of Motion

orbit around the Earth. This change in velocity is caused by the gravitational force exerted by the Earth on the Moon. When a coiled spring is pulled, as in Figure 5.1a, the spring stretches. When a stationary cart is pulled, as in Figure 5.1b, the cart moves. When a football is kicked, as in Figure 5.1c, it is both deformed and set in motion. These situations are all examples of a class of forces called contact forces. That is, they involve physical contact between two objects. Other examples of contact forces are the force exerted by gas molecules on the walls of a container and the force exerted by your feet on the floor. Another class of forces, known as field forces, does not involve physical contact between two objects. These forces act through empty space. The gravitational force of attraction between two objects with mass, illustrated in Figure 5.1d, is an example of this class of force. The gravitational force keeps objects bound to the Earth and the planets in orbit around the Sun. Another common field force is the electric force that one electric charge exerts on another (Fig. 5.1e), such as the attractive electric force between an electron and a proton that form a hydrogen atom. A third example of a field force is the force a bar magnet exerts on a piece of iron (Fig. 5.1f). The distinction between contact forces and field forces is not as sharp as you may have been led to believe by the previous discussion. When examined at the atomic level, all the forces we classify as contact forces turn out to be caused by electric (field) forces of the type illustrated in Figure 5.1e. Nevertheless, in developing mod-els for macroscopic phenomena, it is convenient to use both classifications of forces. The only known fundamental forces in nature are all field forces: (1) gravitational forces between objects, (2) electromagnetic forces between electric charges, (3) strong forces between subatomic particles, and (4) weak forces that arise in certain radioac-tive decay processes. In classical physics, we are concerned only with gravitational and electromagnetic forces. We will discuss strong and weak forces in Chapter 46.

The Vector Nature of ForceIt is possible to use the deformation of a spring to measure force. Suppose a verti-cal force is applied to a spring scale that has a fixed upper end as shown in Fig-ure 5.2a. The spring elongates when the force is applied, and a pointer on the scale reads the extension of the spring. We can calibrate the spring by defining a reference force F

S1 as the force that produces a pointer reading of 1.00 cm. If we

now apply a different downward force FS

2 whose magnitude is twice that of the ref-erence force F

S1 as seen in Figure 5.2b, the pointer moves to 2.00 cm. Figure 5.2c

shows that the combined effect of the two collinear forces is the sum of the effects of the individual forces. Now suppose the two forces are applied simultaneously with F

S1 downward and

FS

2 horizontal as illustrated in Figure 5.2d. In this case, the pointer reads 2.24 cm. The single force F

S that would produce this same reading is the sum of the two vec-

tors FS

1 and FS

2 as described in Figure 5.2d. That is, 0 FS1 0 5 !F12 1 F2

2 5 2.24 units,

b c

M

Field forces

d

!qm "Q

e

Iron N S

f

Contact forces

a

Figure 5.1 Some examples of applied forces. In each case, a force is exerted on the object within the boxed area. Some agent in the environment external to the boxed area exerts a force on the object.

Isaac NewtonEnglish physicist and mathematician (1642–1727)Isaac Newton was one of the most brilliant scientists in history. Before the age of 30, he formulated the basic concepts and laws of mechanics, discovered the law of universal gravita-tion, and invented the mathematical methods of calculus. As a consequence of his theories, Newton was able to explain the motions of the planets, the ebb and flow of the tides, and many special features of the motions of the Moon and the Earth. He also interpreted many fundamental obser-vations concerning the nature of light. His contributions to physical theories dominated scientific thought for two centuries and remain important today.

Brid

gem

an-G

iraud

on/A

rt Re

sour

ce, N

Y

1Serway & Jewett, “Physics for Scientists and Engineers”.

Forces are Vectors

We typically draw forces as vector arrows like this:

5.2 Newton’s First Law and Inertial Frames 113

and its direction is u 5 tan21 (20.500) 5 226.6°. Because forces have been experi-mentally verified to behave as vectors, you must use the rules of vector addition to obtain the net force on an object.

5.2 Newton’s First Law and Inertial FramesWe begin our study of forces by imagining some physical situations involving a puck on a perfectly level air hockey table (Fig. 5.3). You expect that the puck will remain stationary when it is placed gently at rest on the table. Now imagine your air hockey table is located on a train moving with constant velocity along a perfectly smooth track. If the puck is placed on the table, the puck again remains where it is placed. If the train were to accelerate, however, the puck would start moving along the table opposite the direction of the train’s acceleration, just as a set of papers on your dashboard falls onto the floor of your car when you step on the accelerator. As we saw in Section 4.6, a moving object can be observed from any number of reference frames. Newton’s first law of motion, sometimes called the law of inertia, defines a special set of reference frames called inertial frames. This law can be stated as follows:

If an object does not interact with other objects, it is possible to identify a ref-erence frame in which the object has zero acceleration.

Such a reference frame is called an inertial frame of reference. When the puck is on the air hockey table located on the ground, you are observing it from an inertial reference frame; there are no horizontal interactions of the puck with any other objects, and you observe it to have zero acceleration in that direction. When you are on the train moving at constant velocity, you are also observing the puck from an inertial reference frame. Any reference frame that moves with constant veloc-ity relative to an inertial frame is itself an inertial frame. When you and the train accelerate, however, you are observing the puck from a noninertial reference frame because the train is accelerating relative to the inertial reference frame of the Earth’s surface. While the puck appears to be accelerating according to your obser-vations, a reference frame can be identified in which the puck has zero acceleration.

WW Newton’s first law

WW Inertial frame of reference

01234

A downward force elongates the spring 1.00 cm.

F1S

F1S

F1S

01234


F2S

F2S

F2S

01234

When and are applied together in the same direction, the spring elongates by 3.00 cm.

F2S

F1S

When is downward and is horizontal, the combination of the two forces elongates the spring by 2.24 cm.

F2S

F1S

a b

01

23

4

F2S

dc

u

F1S

FS

Figure 5.2 The vector nature of a force is tested with a spring scale.

Airflow

Electric blower

Figure 5.3 On an air hockey table, air blown through holes in the surface allows the puck to move almost without friction. If the table is not accelerating, a puck placed on the table will remain at rest.

1Figure from Serway & Jewett.

Net Force

Net Force

the vector sum of all forces acting on an object.

#»

Fnet =∑i

#»

F i








01234


F1S

F1S

F1S

01234


F2S

F2S

F2S

01234


F2S

F1S


F2S

F1S

a b

01

23

4F2S

dc

u

F1S

FS


Airflow

Electric blower


In the diagram#»

F =#»

F1 +#»

F2.

Net Force








01234


F1S

F1S

F1S

01234


F2S

F2S

F2S

01234


F2S

F1S


F2S

F1S

a b

01

23

4

F2S

dc

u

F1S

FS


Airflow

Electric blower


In the diagram#»

F =#»

F1 +#»

F2.

The magnitude of#»

F is

F =√F 21 + F 2

2 =√

12 + 22 = 2.23 N

The direction of#»

F is

θ = tan−1(F1/F2) = 26.6◦

Net Force Question

A hockey puck is acted on by one or more forces, as shown. Whatis the net force on each puck?

CONCEPTUAL EXERCISES 135

t

vx

Object A

t

vy

t

vx

Object B

t

vy

t

vx

Object C

t

vy

▲ FIGURE 5–18 Conceptual Exercise 3

Conceptual Exercises(Answers to odd-numbered Conceptual Exercises can be found in the back of the book.)

1. A small car collides with a large truck. (a) Is the force experi-enced by the car greater than, less than, or equal to the force ex-perienced by the truck? Explain. (b) Is the acceleration experi-enced by the car greater than, less than, or equal to theacceleration experienced by the truck? Explain.

2. A skateboarder on a ramp is accelerated by a nonzero netforce. For each of the following statements, state whether it isalways true, never true, or sometimes true. (a) The skate-boarder is moving in the direction of the net force. (b) The ac-celeration of the skateboarder is at right angles to the netforce. (c) The acceleration of the skateboarder is in the samedirection as the net force. (d) The skateboarder is instanta-neously at rest.

3. Three objects, A, B, and C, have x and y components of veloc-ity that vary with time as shown in Figure 5–18. What is thedirection of the net force acting on each of these objects, as mea-sured from the positive x axis. (All of the nonzero slopes havethe same magnitude.)

(d) A skydiver parachuting downward with constant speed.(e) A baseball during its flight from pitcher to catcher (ignoringair resistance).

8. An object of mass m is initially at rest. After a force of magni-tude F acts on it for a time T, the object has a speed v. Supposethe mass of the object is doubled, and the magnitude of theforce acting on it is quadrupled. In terms of T, how long does ittake for the object to accelerate from rest to a speed v now?

9. You jump out of an airplane and open your parachute after abrief period of free fall. To decelerate your fall, must the forceexerted on you by the parachute be greater than, less than, orequal to your weight?

10. A hockey puck is acted on by one or more forces, as shown inFigure 5–19. Rank the four cases, A, B, C, and D, in order of themagnitude of the puck’s acceleration, starting with the small-est. Indicate ties with an equal sign.

7 N5 N

A

3 N

3 N

B

3 N3 N

C

3 N

D


F = 3 Nv = 7 m/s

A

F = 3 N

v = 7 m/s

C

F = 3 Nv = 0

B


11. Each of the three identical hockey pucks shown in Figure 5–20 isacted on by a 3-N force. Puck A moves with a speed of 7 m/s in adirection opposite to the force; puck B is instantaneously at rest;puck C moves with a speed of 7 m/s at right angles to the force.Rank the three pucks in order of the magnitude of their accelera-tion, starting with the smallest. Indicate ties with an equal sign.

4. You drop two balls of equal diameter from the same height atthe same time. Ball 1 is made of metal and has a greater massthan ball 2, which is made of wood. If the upward force due toair resistance is the same for both balls, does ball 1 reach theground before, after, or at the same time as ball 2?

5. Riding in an elevator moving upward with constant speed, youbegin a game of darts. Do you have to aim your darts higher,lower, or the same as when you play darts on solid ground?

6. Riding in an elevator moving with a constant upward accelera-tion, you begin a game of darts. Do you have to aim your dartshigher, lower, or the same as when you play darts on solidground?

7. Give the direction of the net force acting on each of the followingobjects. If the net force is zero, state “zero.” (a) A car acceleratingnorthward from a stoplight. (b) A car traveling southward andslowing down. (c)Acar traveling westward with constant speed.

24. Since a bucket of water is “weightless” in space, would it hurtto kick the bucket? Explain.

25. In the movie The Rocketeer, a teenager discovers a jet-poweredbackpack in an old barn. The backpack allows him to fly atincredible speeds. In one scene, however, he uses the backpackto rapidly accelerate an old pickup truck that is being chased by

“bad guys.” He does this by bracing his arms against the cab ofthe pickup and firing the backpack, giving the truck the accel-eration of a drag racer. Is the physics of this scene “Good,”“Bad,” or “Ugly?” Explain.

26. List three common objects that have a weight of approxi-mately 1 N.

WALKMC05_0131536311.QXD 12/8/05 16:43 Page 135

In case C, assume that the forces make an angle of 60◦ to eachother.

1Figure from Walker, “Physics”, page .

Summary

• relative motion and projectiles

• forces

• net force

Test 2 Monday, Feb 24.

HomeworkWalker Physics:

• prev: Ch 4, onward from page 100. Con. Ques: 7, 9;Problems: 1, 40 & 41, 43, 71, 77, 87, 67 (projectile indisguise)

• Read ahead in Ch 5.

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