Kinematics NotesMotion in 1 Dimension

Average Speed and Average Velocity

Average speed describes how fast a particle is moving. It is calculated by:

Average velocity describes how fast the displacement is changing with respect to time:

always positivedistanceaverage speedelapsed time

avexvt

sign gives direction

Average Acceleration Average acceleration describes how fast the velocity is

changing with respect to time. The equation is:

sign determines directionave

xv tat t

Sample problem: A motorist drives north at 20 m/s for 20 km and then continues north at 30 m/s for another 20 km. What is his average velocity?

Sample problem: It takes the motorist one minute to change his speed from 20 m/s to 30 m/s. What is his average acceleration?

Average Velocity from a Graph

t

xA

Bx

t

avexvt

Average Velocity from a Graph

avexvt

t

x ABx

t

Average Acceleration from a Graph

t

vA

Bv

t

avevat

Sample problem: From the graph, determine the average velocity for the particle as it moves from point A to point B.

0-1-2

12

0 0.1 0.2 0.3 0.4 0.5-3

3

t(s)

x(m)

AB

Sample problem: From the graph, determine the average speed for the particle as it moves from point A to point B.

0-1-2

12

0 0.1 0.2 0.3 0.4 0.5-3

3

t(s)

x(m)

AB

Instantaneous Speed, Velocity, and Acceleration

Average Velocity from a Graph

t

x

Remember that the average velocity between the time at A and the time at B is the slope of the connecting line.

AB

Average Velocity from a Graph

t

x

What happens if A and B become closer to each other?

AB

Average Velocity from a Graph

t

x

What happens if A and B become closer to each other?

A B

Average Velocity from a Graph

t

x

AB

What happens if A and B become closer to each other?

Average Velocity from a Graph

t

x

AB

What happens if A and B become closer to each other?

Average Velocity from a Graph

t

x

AB

The line “connecting” A and B is a tangent line to the curve. The velocity at that instant of time is represented by the slope of this tangent line.

A and B are effectively the same point. The time difference is effectively zero.

Sample problem: From the graph, determine the instantaneous speed and instantaneous velocity for the particle at point B.

0-1-2

12

0 0.1 0.2 0.3 0.4 0.5-3

3

t(s)

x(m)

AB

Average and Instantaneous Acceleration

t

v

Average acceleration is represented by the slope of a line connecting two points on a v/t graph.

Instantaneous acceleration is represented by the slope of a tangent to the curve on a v/t graph.

A

B

C

t

x

Instantaneous acceleration is negative where curve is concave down

Instantaneous acceleration is positive where curve is concave up

Instantaneous acceleration is zero where slope is constant

Average and Instantaneous Acceleration

Sample problem: Consider an object that is dropped from rest and reaches terminal velocity during its fall. What

would the v vs t graph look like?

t

v

Sample problem: Consider an object that is dropped from rest and reaches terminal velocity during its fall. What

would the x vs t graph look like?

t

x

Estimate the net change in velocity from 0 s to 4.0 s

a (m/s2)

1.0

t (s)2.0 4.0-1.0

Estimate the net displacement from 0 s to 4.0 s

v (m/s)

2.0

t (s)2.0 4.0

Derivatives

Sample problem. From this position-time graph

x

t

Draw the corresponding velocity-time graph

x

t

Suppose we need instantaneous velocity, but don’t have a graph? Suppose instead, we have a function for the motion of

the particle. Suppose the particle follows motion described by

something like x = (-4 + 3t) m x = (1.0 + 2.0t – ½ 3 t2) m x = -12t3

We could graph the function and take tangent lines to determine the velocity at various points, or…

We can use differential calculus.

Instantaneous Velocity

avexvt

0 0

lim liminst avet t

x dxv vt dt

Mathematically, velocity is referred to as the derivative of position with respect to time.

Instantaneous Acceleration

0 0

lim lim

ave

avet t

vat

v dva at dt

Mathematically, acceleration is referred to as the derivative of velocity with respect to time

Instantaneous Acceleration Acceleration can also be referred to as the second

derivative of position with respect to time.

2

20limt

xd xta

t dt

Just don’t let the new notation scare you; think of the d as a baby , indicating a very tiny change!

Evaluating Polynomial Derivatives It’s actually pretty easy to take a derivative of a

polynomial function. Let’s consider a general function for position, dependent on time.

1

n

n

x Atdxv nAtdt

Sample problem: A particle travels from A to B following the function x(t) = 3.0 – 6t + 3t2.

A) What are the functions for velocity and acceleration as a function of time?

B) What is the instantaneous velocity at 6 seconds?

C) What is the initial velocity?

Sample problem: A particle travels from A to B following the function x(t) = 2.0 – 4t + 3t2 – t3.

a) What are the functions for velocity and acceleration as a function of time?

b) What is the instantaneous acceleration at 6 seconds?

Sample problem: A particle follows the function2

4.21.5 5x tt

a) Find the velocity and acceleration functions.

b) Find the instantaneous velocity and acceleration at 2.0 seconds.

Kinematic Equation Review

Here are our old friends, the kinematic equations

212

2 20 2 ( )

o

o o

v v at

x x v t at

v v a x

Sample problem (basic): Show how to derive the 1st kinematic equation from the 2nd.

Sample problem (advanced): Given a constant acceleration of a, derive the first two kinematic equations.

Draw representative graphs for a particle which is stationary.

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

Draw representative graphs for a particle which has constant non-zero velocity.

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

Draw representative graphs for a particle which has constant non-zero acceleration.

Sample problem: A body moving with uniform acceleration has a velocity of 12.0 cm/s in the positive x direction when its x coordinate is 3.0 cm. If the x coordinate 2.00 s later is -5.00

cm, what is the magnitude of the acceleration?

Sample problem: A jet plane lands with a speed of 100 m/s and can accelerate at a maximum rate of -5.00 m/s2 as it comes to a halt.

a) What is the minimum time it needs after it touches down before it comes to a rest?

b) Can this plane land at a small tropical island airport where the runway is 0.800 km long?

Freefall

Free Fall Free fall is a term we use to indicate that an

object is falling under the influence of gravity, with gravity being the only force on the object.

Gravity accelerates the object toward the earth the entire time it rises, and the entire time it falls.

The acceleration due to gravity near the surface of the earth has a magnitude of 9.8 m/s2. The direction of this acceleration is DOWN.

Air resistance is ignored.

Sample problem: A student tosses her keys vertically to a friend in a window 4.0 m above. The keys are caught 1.50

seconds later. a) With what initial velocity were the keys tossed?

b) What was the velocity of the keys just before they were caught?

Sample problem: A ball is thrown directly downward with an initial speed of 8.00 m/s from a height of 30.0 m. How many seconds later does the ball strike the ground?