Ch 2 and 4

Copyright © 2012 Pearson Education Inc.

CHAPTERS 2 AND 4

MOTION IN ONE AND TWO DIMENSIONS


Goals for Chapter 2 • To describe straight-line motion in terms of velocity and

acceleration

• To distinguish between average and instantaneous velocity and average and instantaneous acceleration

• To interpret graphs of position versus time, velocity versus time, and acceleration versus time for straight-line motion

• To understand straight-line motion with constant acceleration

• To examine freely falling bodies


In order to solve a kinematics problem, you must set up a coordinate system – define an origin and a positive direction.

Position, Distance and Displacement

Distance: the total length of travel; if you drive from your house to the grocery store and back, you have covered a distance of 8.6 mi.

Displacement: the change in position. If you drive from your house to the grocery store and then to your friend’s house, your displacement is (-) 2.1 mi and the distance you have traveled is 10.7 mi.


Position, Distance and Displacement

Example: You walk from your house to the library and then to the park. What is the (a) distance traveled and (b) displacement?

Example: If you walk from library to your house and then to the park, what is the (a) distance traveled and (b) displacement?


Average Velocity

• The change in the particle’s coordinate is ∆x = x2 − x1.

• The average x-velocity of the particle is vav-x = ∆x/∆t.


Negative Velocity

• The average x-velocity is negative during a time interval if the particle moves in the negative x-direction for that time interval.


A Position-Time (x-t) Graph

A position-time graph (an x-t graph) shows how the average x-velocity is related to the slope of an x-t graph.


Finding Instantaneous Velocity on an x-t Graph

At any point on an x-t graph, the instantaneous x-velocity is equal to the slope of the tangent to the curve at that point.


Motion Diagrams • A motion diagram shows the position of a particle at various instants, and

arrows represent its velocity at each instant.

• Figure shows the x-t graph and the motion diagram for a moving particle.


Average acceleration • Acceleration describes the rate of change of velocity with time.

• The average x-acceleration is aav-x = ∆vx/∆t.

An astronaut, attached to a maneuvering unit, has velocities recorded over 2 sec time durations as shown in the figure. Calculate the avg. acceleration for each 2 sec duration and state whether the speed increases or decreases over each of those?


Finding acceleration on a vx-t graph

• As shown in Figure, the υx-t graph may be used to find the instantaneous acceleration and the average acceleration.


A vx-t graph and a motion diagram

• Figure shows the vx-t graph and the motion diagram for a particle.


Motion with constant acceleration

• For a particle with constant acceleration, the velocity changes at the same rate throughout the motion.


Two bodies with different accelerations

Example: A motorist travelling with a constant speed of 15 m/s passes a school crossing where speed limit is 10 m/s. At the school-crossing sign a policer office, at rest, starts pursuing the motor car with a constant acceleration of 3.0 m/s2. (a) How much time elapses before the officer crosses the motorist? At that time (b) what is the officer’s speed and (c) how far has each vehicle traveled?


The Equations of Motion with Constant Acceleration

Example 2.14: A lightning is observed in the sky. 3.5 sec later a thunder is heard. If the speed of sound is 343 m/s, how far away from the observer was the lightning bolt?

Example 2.25: The position of a particle as a function of time is given by x = (6 m/s)t + ½ (-2 m/s2)t2. Plot x vs t for o to 2 sec. Also, find (a) average speed and (b) average velocity, from t = 0 to 1 sec.

Example 2.55: A 27 pound meteorite struck a car leaving a dent of 22 cm. If the meteorite struck the car with a speed of 130 m/s, what was the magnitude of its acceleration?


• Free fall is the motion of an object under the influence of only gravity.

• In the figure, a strobe light flashes with equal time intervals between flashes.

• The velocity change is the same in each time interval, so the acceleration is constant.

Freely Falling Objects


Freely Falling Objects • Aristotle thought that heavy bodies fall faster than light ones, but

Galileo showed that all bodies fall at the same rate. • If there is no air resistance, the downward acceleration of any freely

falling object is g = 9.8 m/s2 = 32 ft/s2.

VIDEO


Up-and-Down Motion in Free Fall An object is in free fall even when it is moving upward.

Example 2.88: You shoot an arrow into the air. Two seconds later the arrow has gone straight upwards to a height of 30.0 m above the launch point. (a) What was the arrow’s initial speed? (b) How long did it take for the arrow to first reach a height of 15 m. from above the launch point?


Clicker Quiz on Chapter 2


Position vector

• The position vector from the origin to point P has components x, y, and z.


Average velocity

• The average velocity between two points is the displacement divided by the time interval between the two points, and it has the same direction as the displacement.


Instantaneous velocity

• The instantaneous velocity is the instantaneous rate of change of position vector with respect to time.

• The components of the instantaneous velocity are vx = dx/dt, vy = dy/dt, and vz = dz/dt.

• The instantaneous velocity of a particle is always tangent to its path.


• A frame of reference is a coordinate system plus a time scale. • Two frames of references: Train (T) and Ground (G) • Observer is on the ground while a person is on the train • Velocity of the person with respect to the train: vPT (along x-direction) • Velocity of the train with respect to the ground: vTG (along x-direction) • Combining, velocity of the person with respect to the ground: vPG = vPT + vTG

Relative Velocity: One Dimension

vPG = vPT + vTG = 1.2 + 15 = 16.2 m vPG = vPT + vTG = -1.2 + 15 = 13.8 m


You are driving north at a constant speed of 88 km/h. A truck is traveling in the opposite lane, south bound, at 104 km/h. (a) What is the truck’s velocity relative to you, (b) what is your velocity with respect to the truck and (c) how do the relative velocities change after passing?

Relative Velocity: One Dimension


Relative Velocity: Two Dimensions

vPG = √(vPT2 + vTG

2) = √(1.22 + 152) = 15.05 m Direction of vPG = tan-1 (vPT/vTG) = 4.6º.

• Velocity of the person with respect to the train: vPT (along y-direction) • Velocity of the train with respect to the ground: vTG (along x-direction)


Part 1

The compass of an airplane indicates that it is headed due north and its airspeed indicator shows that it is moving through the air at 240 km/h. if there is a wind of 100 km/h from west to east, what is the velocity of the airplane with respect to the ground?

Relative Velocity: Two Dimensions

Part 2

In what direction should the pilot head to travel due north? What will be the velocity of plane relative to the earth? Assume the same airspeed and the velocity of the wind are the same as part 1.


Calculating Average and Instantaneous Velocity The co-ordinates of a Mar rover’s position are given by, x = 2 (m) – 0.25 (m/s2)t2, y = 1 (m/s)t + 0.025 (m/s3)t3. Using the coordinates at 0 and 2 sec, find the displacement and avg. velocity vectors during the interval from 0 to 2 sec. Derive instantaneous velocity vector to calculate velocity at 2 sec.

θ


The co-ordinates of a Mar rover’s position are given by, x = 2 (m) – 0.25 (m/s2)t2, y = 1 (m/s)t + 0.025 (m/s3)t3. Using the coordinates at 0 and 2 sec, find the displacement and avg. velocity vectors during the interval from 0 to 2 sec. Derive instantaneous velocity vector to calculate velocity at 2 sec.

Calculating Average and Instantaneous Velocity


Average Acceleration

• The average acceleration during a time interval ∆t is defined as the velocity change during ∆t divided by ∆t.


Instantaneous Acceleration

• The instantaneous acceleration is the instantaneous rate of change of the velocity with respect to time.

• Any particle following a curved path is accelerating, even if it has constant speed.

• The components of the instantaneous acceleration are ax = dvx/dt, ay = dvy/dt, and az = dvz/dt.


Calculating Average and Instantaneous Acceleration

Return to the Mars rover example. Calculate average acceleration between 0 to 2 sec and instantaneous acceleration at 2 sec.


Two Dimensional Motion Example 4.3: You are walking with a constant speed of 1.75 m/s, in a direction, 18 deg north of east. How much time does it take to change your displacement by (a) 20 m east or (b) 30 m north?

Example 4.4: Starting from rest a car accelerates at 2 m/s2 up a hill that is inclined 5.5 deg above the horizontal. How far horizontally and vertically has the car traveled in 12 s?


Projectile Motion A projectile is any body given an initial velocity that then follows a path determined by the effects of gravity


The x and y Motions are Separable

• The red ball is dropped at the same time that the yellow ball is fired horizontally.

• The strobe marks equal time intervals.

• We can analyze projectile motion as horizontal motion with constant velocity and vertical motion with constant acceleration: ax = 0 and ay = −g.


The Symmetry of Projectile Motion • The trajectory is a symmetric parabola • Horizontally: equal distances in equal time intervals, vx is constant (ax = 0 m/s2) • Vertically: Velocity changes by equal amounts in equal time intervals (ay = 9.8 m/s2)


Projectile Motion

Example 4.6: An electron in a cathode ray tube is moving horizontally at 2 x 109 cm/s when deflection plate gives it an upward acceleration of 5.3 x 1017 cm/s2. (a) How long does it take for the electron to cover a horizontal distance of 6.2 cm? (b) What is its vertical displacement during this time?

Example 4.25: A ball rolls off a table and falls 0.75 m to the floor, landing with a speed of 4 m/s. (a) What is the acceleration of the ball just before it strikes the ground, (b) what was the initial speed of the ball and (c) what initial speed must the ball have if it is to land with a speed of 5.0 m/s?

Example 4.35: Snowballs are thrown with a speed of 13 m/s from a roof 7 m above the ground. Snowball A is thrown straight downward, while snowball B is thrown in a direction 25 deg above the horizontal. (a) How does the landing speed of snowball A compare to B?


Projectile Motion


Uniform Circular Motion

• For uniform circular motion, the speed is constant and the acceleration is perpendicular to the velocity.


Motion With Constant Perpendicular Acceleration


Acceleration for Uniform Circular Motion

• For uniform circular motion, the instantaneous acceleration always points toward the center of the circle and is called the centripetal acceleration.

• The magnitude of the acceleration is arad = v2/R.

• The period T is the time for one revolution, and arad = 4π2R/T2.


Acceleration for Uniform Circular Motion

Example 4.21: A Ferris wheel with a radius of 5.00 m completes one revolution every 32 sec. (a) What is the average speed of the rider on this ferris wheel, (b) what is the radial (centripetal) acceleration acting on the passengers and (c) if a rider accidentally drops a stuffed animal at the top of the wheel, where does it land relative to the base of the ride?

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Ch 2 and 4

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