Motion

Measuring Motion

• Speed– Average Speed = distance covered / time taken

v = d/t• metric unit of speed: m/s• English unit of speed: ft/s

– Constant speed:• moving equal distances in equal time periods• an object covering 5 feet each second has a constant speed of

5ft/s

– If speed changes:• Average speed: average over all speeds • Instantaneous speed: speed at any given instant

Constant speed:this car is moving in a straight line covering a distance of 1 mi each minute. The car, therefore, has a constant speed of 60 mi per each 60 min, or 60 mi/hr.

Speed is the slope of the straight line graph of distance (on the y-axis) versus time (on the x-

axis)

Velocity• shows how fast and in what direction an

object moves • Velocity: speed + direction of motion• it is a vector quantity• vector: a quantity that has both magnitude

(size) and direction– ex.: velocity, acceleration, force, etc.

• scalar: a quantity without direction (has only magnitude)– ex.: speed, time, distance, volume, surface area,

etc.

Velocity is a vector that we can represent graphically with arrows. Here are three different velocities represented by three different arrows. The length of each arrow is proportional to the speed and the arrowhead shows the direction of travel.

Acceleration– Three ways to change motion:

• change speed• change direction• change both speed and direction at the same time

– Average acceleration: change in velocity over the time taken to make the change

vf – vi vi = initial velocity

t vf = final velocity

t = time interval

– Metric unit for acceleration:unit of velocity m/s

unit of time s – English unit for acceleration: ft/s2

a =

= = m/s2

Four different ways (A-D) to accelerate a car.

30 mi/hr 60 mi/hr60 mi/hr 60 mi/hr 30 mi/hr

(A) This graph shows how the speed changes per unit of time while driving at a constant 30 mi/hr in a straight line. As you

can see, the speed is constant, and for straight-line motion, the acceleration is 0.

(B) This graph shows the speed increasing to 50 mi/hr when moving in a straight line for 5 s. The acceleration is the slope of the straight line graph of speed (on the y-axis) versus time (on the x-axis).

Forces

• result from two kinds of interactions:– contact interactions– interaction at a distance (ex.: gravitational force)

• force: a push or a pull – changes the motion of an object– it is a vector: has both magnitude and direction

Graphical Representation of a Force

• represented by an arrow– the tail of a force arrow is placed on the object

that “feels” the force– the arrowhead points in the direction of the

applied force– the length of the arrow is proportional to the

magnitude of the applied force

forceobject

Adding Forces

• net force (resultant): the sum of all forces acting on an object

• when two or more forces act on an object their effects are cumulative (added together)

• forces are added considering:– their directions– their magnitudes (sizes)

• the net force (resultant) can be calculated using geometry

(A)When two parallel forces are acting on the cart in the same direction, the net force is the two forces added together.

10 lb west

• (B) When two forces are opposite and of equal magnitude, the net force is zero.

10 lb west

• (C) When two parallel forces are not of equal magnitude, the net force is the difference and the direction is that of the larger force.

10 lb west

Adding nonparallel forces

• To add two forces that are not parallel:– draw the two force vectors to scale– place the tip of one of the forces at the tail of the other– draw a vector to close the triangle: this is the net force

(the vector sum of the two forces)

F1

F2

vector sum

(A) This shows two equal forces (200 N each) acting at an angle of 90O, which give a resultant force (Fnet) of 280 N acting at 45O. (B) Two unequal forces acting at an angle of 60O give a single

resultant of about 140 N.

Galileo (1564-1642) challenged the

Aristotelian view of motion and focused

attention on the concepts of distance, time, velocity, and

acceleration

Inertia

• Galileo:– natural tendency of objects:

• at rest• in motion: explained the behavior of matter to

stay in motion by inertia

– inertia: natural tendency of an object to remain at rest or in unchanged motion in the absence of any forces

(A) This ball is rolling to your left with no forces in the direction of motion. The vector sum of the force of floor friction (Ffloor) and the force of air friction (Fair) result in a net force opposing the

motion, so the ball slows to a stop.

(B) A force is applied to the moving ball, perhaps by a hand that moves along with the ball. The force applied (Fapplied) equals the vector sum of the forces opposing the motion, so the ball continues to move with a constant velocity.

Thus, an object moving through space without any opposing friction (A) continues in a straight line path at a constant speed.

The application of an unbalanced force as shown by the large

arrow, is needed to (B) slow down, (C) speed up, or, (D) change the

direction of travel.

Falling Objects• free fall : due to force of gravity on the object

• the velocity of a falling object does not depend on its mass

• in the absence of air resistance (in a vacuum) all objects fall at the same velocity– differences in the velocities of falling objects are due to

air resistance

According to a widespread story, Galileo dropped two

objects with different weights from the Leaning Tower of Pisa. They were supposed to have hit the ground at about the same

time, discrediting Aristotle's view that the speed during the fall is proportional to weight.

Acceleration Due to Gravity: g = 9.8 m/s2 = 32 ft/s2

• Free Fall: – at a constant acceleration caused by the force

of gravity– all objects experience this constant acceleration– this acceleration is 9.8 m/s2 or 32 ft/s2

– This means that the velocity of a free falling object increases at a constant rate (i.e., by 9.8 m/s every one second, or by 32 ft/s every one second)

The velocity of a falling object increases at a constant rate (i.e., by 32 ft/s each second)

– remember the equation for velocity:v = d/ t

– can be rearranged to incorporate acceleration, distance, and time.• solve for distance:

d = vt– an object in free fall has uniform (constant) acceleration,

so we can calculate the average velocity as:

vi + vf 2

– substitute this equation into d = vt to get

(vi + vf) (t) 2

v =

d =

• vi = 0 (for a free falling object)

(vf) (t) 2

• use the acceleration equation:

vf - vi t

but vi = 0 , which gives a = vf /t

Solve for vf : vf = at

• Substituting vf in the equation for d above, we get:

(at) (t) 2

d = (1/2)at2 which gives the distance covered in the free fall

d =

a =

d =

An object dropped from a tall building covers increasing distances with every successive second of falling. The distance covered is proportional to the square of the time falling (d t2).

Projectile motion• when an object is thrown into the air by a

given force

• projectile motion can be:• straight up vertically

• object thrown straight out horizontally

• object thrown at some angle in between these two

• in a projectile motion:• gravity acts on objects at all times (regardless of

their position)

• acceleration due to gravity is constant and independent of the motion of the object

Vertical Projectiles

• an object thrown straight up into the air • gravity acts on the object at all times,

pulling it down– as the object moves up its velocity decreases

(gravitational force slows down the object)– at the peak of the ascent, the object comes to

rest (for an instant) and begins its fall toward the Earth

– during the fall its velocity increases at a constant rate (i.e., acceleration is constant and equal to g, acceleration due to gravity).

On its way up, a vertical projectile such as a

misdirected golf ball is slowed by the force of

gravity until an instantaneous stop; then it

accelerates back to the surface, just as another

golf ball does when dropped from the same

height. The straight up and down moving golf ball has been moved to the side in the sketch so we can see

more clearly what is happening.

Horizontal Projectiles

• an object thrown straight out horizontally– the force of gravity acts on the object at all

times, pulling it down

• the motion of the object can be broken down into two components:– a horizontal motion at constant velocity– a vertical motion with constant acceleration

(i.e., increasing velocity)

A horizontal projectile has a constant horizontal velocity and an increasing vertical velocity as it falls to the ground. The combined effect of the two velocities results in a curved path (parabola). Neglecting air resistance, an arrow shot horizontally will strike the ground at the same time as one dropped from the same height above the ground, as shown here by the increasing vertical velocity arrows.

Projectile thrown at an angle

A football is thrown at some angle. Neglecting air resistance, the horizontal velocity is constant, and the vertical velocity decreases (on the way up) then increases (on the way down), just as in the case of a vertical projectile. The combined motion produces a parabolic path.