1D Motion

LECTURE 2 1

Reading 2-1 to 2-4

One dimensional motion◦ Displacement and Distance

◦ Velocity and Speed

◦ Acceleration

◦ Motion with a Constant Acceleration

LC 2-1 questionA student goes from 𝑥1 = 2m to 𝑥2 = 12m to 𝑥3 = 5m. What is there their total displacement in meters?

LECTURE 2 2

x [m]0 5 10

x1 x2x3

LC 2-2 questionA student goes from 𝑥1 = 2m to 𝑥2 = 12m to 𝑥3 = 5m. What is there their total distance traveled in meters?

LECTURE 2 4

x [m]0 5 10

x1 x2x3

LC 2-3 questionIt takes 2.0s to travel from 𝑥1 to 𝑥2 and 3.0s to travel from 𝑥2 to 𝑥3. What is the average velocity in m/s?

LECTURE 2 6

x [m]0 5 10

x1 x2x3

2.0s

3.0s

LC 2-4 questionIt takes 2.0s to travel from 𝑥1 to 𝑥2 and 3.0s 𝑥2 to 𝑥3. What is the average speed?

LECTURE 2 8

x [m]0 5 10

x1 x2x3

2.0s

3.0s

Instantaneous Velocity and SpeedInstantaneous velocity in the 𝑥direction is defined to be

𝑣𝑥 𝑡 = lim∆𝑡→0

∆𝑥

∆𝑡=𝑑𝑥

𝑑𝑡

Instantaneous speed is the magnitude of the instantaneous velocity.

The slope of the line connecting 2 points represents the average velocity.

Instantaneous velocity is the slope of the tangent line.

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AccelerationAcceleration is a measure of the rate of change in velocity.

Average acceleration in the x-direction is defined to be

𝑎𝑎𝑣𝑒 𝑥 =∆𝑣𝑥∆𝑡

Instantaneous acceleration in the 𝑥 direction is defined to be

𝑎𝑥 𝑡 = lim∆𝑡→0

∆𝑣𝑥∆𝑡

=𝑑𝑣𝑥𝑑𝑡

=𝑑2𝑥

𝑑𝑡2

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LC 2-5 questionThe diagram shows an objects 𝑥-position as a function of time. At which point or points is its instantaneous acceleration zero?

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LC 2-6 questionOn the provided axes, draw a possible curve of 𝑣𝑥 as a function of time for an object that has constant positive acceleration and negative velocity.

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Speeding up or slowing down?When the direction of velocity and acceleration are the same, the object is speeding up.

When the direction of velocity and acceleration are opposite, the object is slowing down.

An object with a negative acceleration is speeding up if its velocity is also negative!

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Constant AccelerationFor an object undergoing constant acceleration, 𝑎𝑥, which is initially at position, 𝑥0, and moving with velocity, 𝑣0𝑥

The velocity at a later time 𝑡 is given by

𝑣𝑥 𝑡 = 𝑎𝑥𝑡 + 𝑣0𝑥

The position at a later time 𝑡 is given by

𝑥 𝑡 = 12𝑎𝑥𝑡

2 + 𝑣0𝑥𝑡 + 𝑥0

Eliminating 𝑡 from the above equations we obtain

𝑣𝑥2 = 𝑣0𝑥

2 + 2𝑎𝑥 𝑥 − 𝑥0

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CheckPoint: Rolling Down a RampAt 𝑡 = 0 a ball, initially at rest, starts to roll down a ramp with constant acceleration. Suppose it moves 1 foot between 𝑡 = 0s and 𝑡 = 1s. How far does it move between 𝑡 = 1s and 𝑡 = 2s?

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Demo 1 – Incline with Flash LightsDemonstration of the distance and the velocity formula under a constant acceleration

𝑥 𝑡 = 12𝑎𝑥𝑡

2 + 𝑣0𝑥𝑡 + 𝑥0

𝑣𝑥 𝑡 = 𝑎𝑥𝑡 + 𝑣0𝑥

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A motor cycle is moving at 30.0m/s when the rider applies the brakes, giving the motorcycle a constant deceleration. During the 3.0s interval immediately after braking begins, the speed decreases to 15.0m/s. What distance does the motorcycle travel from the instant braking begins until it comes to rest?

Example 1

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LC 2-7A Guinea and a feather are dropped from the same height at the same time. Assuming there is no air resistance, which hits the ground first?

LECTURE 2 21

"Guinea 641642" by Classical Numismatic Group, Inc.

http://www.cngcoins.com. Licensed under CC BY-SA 3.0 via

Wikimedia Commons -

http://commons.wikimedia.org/wiki/File:Guinea_641642.jpg#/med

ia/File:Guinea_641642.jpg

Demo 2 – “Guinea and Feather”The acceleration due to gravity near the surface of Earth is constant at 𝑔 = 9.81m/s2 towards the center of the earth.

Neglecting air resistance, how far objects initially at rest fall depends only on time.

LECTURE 2 22

If a particle’s position is given by 𝑥 = 4 − 12𝑡 + 3𝑡2 (where 𝑡 is in seconds, and 𝑥 is in meters)

a) What is its velocity at 𝑡 = 1s?

b) Is it moving toward increasing or decreasing x at 𝑡 = 1s?

c) Is there ever an instant when the velocity is zero?

Example 2

LECTURE 2 23