Chapter 2 : Motion along a Straight Line

Kinematics vs. Dynamics

n  Kinematics is the branch of classical mechanics that describes the motion of bodies (objects) and systems (groups of objects) without consideration of the forces that cause the motion.

à Chapter 2 & 3

n  In contrast, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion.

à Using the dynamics the acceleration can be found. Then, we can use the kinematics to predict the motion

1. Displacement

n  Position : the location of an object

n  Displacement : the direction and distance of the shortest path between an initial and final position. defined as the change in position during some time interval. q  Represented as Δx

q  SI units are meters (m) q  Δx can be positive or negative

n  Different from distance q  Distance is the length of a path followed by a particle.

x, xi , x f

Δx = x f − xi

Distance vs. Displacement – An Example n  Assume a player moves from one end of the court to the other and back. n  Distance is twice the length of the court

q  Distance is always positive n  Displacement is zero

q  Δx = xf – xi = 0 since xf = xi

n  Average velocity : displacement divided by elapsed time.

unit: m/s

vave =

x2 − x1

t2 − t1=ΔxΔt

Position-Time Graph

n The position-time graph shows the motion of the particle (car). n The smooth curve is a guess as to what happened between the data points.

Section 2.1

n  Opposite direction of motion

n  Graphical analysis

average velocity = slope between two points

2. Instantaneous Velocity

vx = lim

Δt→0

ΔxΔt

=dxdt

3. Average and Instantaneous Acceleration

n  Acceleration : change in velocity, unit m/s2

n  Average acceleration

aave =

v2 − v1

t2 − t1=ΔvΔt

n  Instantaneous acceleration

a = lim

Δt→0

ΔvΔt

=dvdt

=d 2xdt2

Motion Diagram

n  Motion diagram q  It represents the motion of an object by displaying its location at

various equally spaced times on the same diagram. q  Motion diagrams are a pictorial description of an object's motion. q  It shows an object's position and velocity initially and presents

acceleration.

n  An object with images that have increasing distance between them is speeding up.

n  Example: A sprinter starting the 100 meter dash.

n  An object with images that have decreasing distance between them is slowing down.

n  Example: A car stopping for a red light.

Examples of Motion Diagram

n  A motion diagram can show more complex motion in two dimensions.

n  Example: A jump shot from center court. n  In this case the ball is

slowing down as it rises, and speeding up as it falls.

Examples of Motion Diagram

4. Motion with Constant Acceleration

n  Motion with constant positive acceleration results in steadily increasing velocity.

n  Equations for constant acceleration motion can be obtained by integration.

vx (t) = v0x + axt

x(t) = x0 + v0xt +12

axt2

vx2 − v0x

2 = 2ax (x − x0 )

x = x0 +12

v0x + vx( )t

Graphs of the motion with constant acceleration

• Example 1) The figure shows a plot of vx(t) for a car traveling in a straight line.

a) What is ! between t = 6 s and t = 11 s?

b) How far does the car travel from time t = 10 s to time t = 15 s?

c) What is ! between t = 6 s and t = 11 s?

d) What is ! for the interval t = 0 to t = 20 s?

ax,av

vx,av

vx,av

• Example 2) A motorist traveling with constant velocity of 16.0 m/s passes a school cross corner, where the speed limit is 11.2 m/s. Just as he passes, a police officer on a motorcycle stopped at the corner starts off in pursuit 2.00 s later with a constant acceleration of 3.00 m/s2.

a) How long will it take to catch up?

b) What is the officer’s speed at that point?

c) What is the total distance the police officer has traveled at that point?

• Example 3) The engineer of a passenger train traveling at 25.0 m/s sights a freight train whose caboose is 200 m ahead on the same track. The freight train is traveling at 15.0 m/s in the same direction as the passenger train. The engineer of the passenger train immediately applies the brakes, causing a constant acceleration of −0.100 m/s2, while the freight train continues with constant speed. Take x=0 at the location of the front of the passenger train when the engineer applies the brakes. Where will a collision take place?

• Example 4) An antelope moving with constant acceleration covers two points 70.0 m apart in 7.00 s. Its speed at the 2nd point is 15.0 m/s.

a) What is its speed at 1st point?

b) What is the acceleration?

•

5. Free Falling Bodies

n  Constant acceleration of free fall: 9.80 m/s2 due to the gravity This number is true near the earth’s surface.

n  Penny and feather

n  Galileo’s experiment on the moon.

• Example 5) You throw a ball vertically upward from the roof of a tall building. The ball leaves your hand at a point even with roof railing with an upward speed of 15.0 m/s; the ball is then in free fall. On its way back down, it just misses the railing. Find

a) the position and velocity of the ball at 1.00 s and 4.00 s after leaving your hand,

n  Equations for the free fall ( x à y )

vy (t) = v0 y + ayt

y(t) = y0 + v0 yt +12

ayt2

vy2 − v0 y

2 = 2ay ( y − y0 )

y = y0 +12

v0 y + vy( )t

b) the velocity when the ball is 5.00 m above the railing,

c) the maximum height reached and time at maximum.

• Example 6) You are on the roof of the physics building, 46.0 m above the ground. Your physics professor, who is 1.80 m tall, is walking alongside the building at a constant speed of 1.20 m/s. If you wish to drop an egg on your professor's head, how far from the building should the professor be when you release the egg?

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