Kinematics and One Dimensional Motion

8.01 W02D1

Next Reading Assignment: W02D2

Young and Freedman: 3.1-3.3, 3.5

Strategy: One Dimensional Kinematics

1.  Define an appropriate coordinate system

2.  Integrate or differentiate

A coordinate system consists of:

1.  An origin at a particular point in space 2.  A set of coordinate axes with scales and labels 3.  Choice of positive direction for each axis with a set of unit

vectors 4.  Choice of type: Cartesian or Polar or Spherical

Example: Cartesian One-Dimensional Coordinate System

Concept Question: Displacement

An object goes from one point in space to another. After the object arrives at its destination, the magnitude of its displacement is:

1) either greater than or equal to

2) always greater than

3) always equal to

4) either smaller than or equal to

5) always smaller than

6) either smaller or larger than

the distance it traveled.

CQ: Representing Motion – 1D Which of these is a possible graph of the position, x(t)?

CQ: Representing Motion – 1D Which of these is a possible graph of the x-component of velocity?

Worked Example: Model Rocket

A person launches a home-built model rocket straight up into the air at y = 0 from rest at time t = 0 . (The positive y-direction is upwards). The fuel burns out at t = t0. The position of the rocket is given by

with a0 and g are positive. Find the y-components of the

velocity and acceleration of the rocket as a function of time. Graph ay vs t for 0 < t < t0.

2 6 400 0 0

1 ( ) / ; 02 30

ay a g t t t t t⎧= − − < <⎨⎩

Table Problem: Sports Car At t = 0 , a sports car starting at rest at x = 0 accelerates

with an x-component of acceleration given by

ax(t) = α t –β t 3, for 0 < t < (α/β)1/2.

and zero afterwards with α, β > 0.

(1)  Find expressions for the velocity and position vectors of the sports as functions of time for t >0.

(2) Sketch graphs of the x-component of the position, velocity and acceleration of the sports car as a function of time for

t >0

Concept Question: One Dimensional Kinematics

The graph shows the position as a function of time for two trains running on parallel tracks. For times greater than t =0, which is true:

1.  At time tB, both trains have the same velocity.

2.  Both trains speed up all the time. 3.  Both trains have the same velocity

at some time before tB, . 4.  Somewhere on the graph, both

trains have the same acceleration.

Worked Example: Runner

A runner accelerates from rest with a constant x-component of acceleration

2.0 m s-2 for 2.0 s and then travels at a constant velocity for an additional 6.0 s. How far did the runner travel?

Concept Question: One Dimensional Kinematics A person standing at the edge of a cliff

throws one ball straight up and another ball straight down, each at the same initial speed. Neglecting air resistance, which ball hits the ground below the cliff with the greater speed:

1.  ball initially thrown upward;

2.  ball initially thrown downward;

3.  neither; they both hit at the same speed.

Concept Question

Table Problem: One Dimensional Kinematics

A motorist traveling with constant speed of 15 m/s passes a school-crossing corner, where the speed limit is 10 m/s. Just as the motorist passes, a police officer on a motorcycle stopped at the corner (x = 0) starts off in pursuit. The officer accelerates from rest at ax = 2.5 m/s2 until reaching a speed of 20 m/s. The officer then slows down at a constant rate until coming alongside the car at x = 360 m, traveling with the same speed as the car. a) How long does it take for the officer to catch up with the motorist? b) How long does officer speed up? c) How far is the officer from the corner and from the car when switching from speeding up to slowing down? d) What is the acceleration of the officer when slowing down? e) Draw an x-t graph and a vx-t graph for the two vehicles.