Advanced Higher Physics Unit 1

Kinematics relationships and relativistic motion

Kinematic relationships

Kinematics is the study of motion without reference to cause.

From Higher Physics, we know:

asuv

at

2

2

1uts

atuv

22

2

We now need to prove this equations using calculus.

VelocityAverage Velocity is defined as the change in displacement (Δs) over time (Δt).

t

svav

Instantaneous velocity is defined as the speed at any particular time during a journey.

This velocity can be found by measuring the average velocity overa very short time interval.

t

sv

lim (as Δt→0)

dt

dsv

AccelerationAcceleration is defined as the change in velocity (Δv) over time (Δt).

t

vaav

Instantaneous acceleration is defined as the acceleration at any particular time during a journey.

This acceleration can be found by measuring the acceleration over a very short time interval.

t

va

lim (as Δt→0)

dt

dva

dt

dsv

dt

dv

dt

ds

dt

d2

2

dt

sd

2

2

dt

sd

dt

dva

If

then

(This formula can be found in the data booklet)

v = u + at

adt

dv

adtdv

catv at t=0, v = u, c=u

atuv

integrate

s = ut +½at²

dtatuds

catuts 2

2

1at t=0, s=0, c=0

atuv

integrate

2

2

1atuts

atudt

ds

v²=u²+2asatuv

2atuv2

atuatuv2 2222 ta2uatuv

taking a common factor of 2a gives

222 at

21

ut2auv

and since s = ut + ½at2

2asuv 22

A very useful extra equation is

tvu

s

2

Displacement-time and velocity-time graphs can be used to derive information.

Example Q1a) 20071. (a) A particle has displacement

s = 0 at time t = 0 and moves with a constant acceleration a.

The velocity of the object is given by the equation v = u + at, where the symbols have their usual meanings.

Using calculus, derive an equation for the displacement s of the object as a function of time t.

Answers

dtatuds

catuts 2

2

1at t=0, s=0, c=0

atuv

integrate

2

2

1atuts

atudt

ds

RelativityThe greatest possible speed is the speed of light in a vacuum:

18100.3 msc

At very high velocities an object appears to have gained mass

to the viewpoint of a stationary observer.

The mass gain can be calculated using:

Available on DATA SHEET

2

2

0

1cv

mm

Rest mass (kg)

speed of object (msˉ¹)

speed of light in a vacuum (msˉ¹)

Available in DATA BOOKLET

Object mass (kg)

The rest mass of an object is its mass when it is stationary.

Relativistic energy

2mcE

The total energy of an object is:

speed of light

(3x108 ms-1)relativistic mass (kg)

relativistic energy (J)

This energy is made of two parts:

Ekcmmc 20

2

Kinetic energy of motionrest mass

energy

relativistic energy

Relativistic effects are only taken into account when v>10%c.

in DATA BOOKLET

Not in DATA BOOKLET

Example Q1. (b) 2007

A proton is accelerated to a high speed so that its mass is 2.8 times its rest mass.

1. Calculate the speed of the proton.

2. Calculate the relativistic energy of a proton at this velocity.

Answers

2

2

0

1cv

mm

m

m

c

v 02

2

1

8.2

1

)103(1

28

2

v

1.Find the speed using: Rearrange

228

2

8.2

1

1031

v

228

2

8.2

11

)103(

v

2

282

8.2

11103v

18108.2 msv

The relativistic energy is given by:

2mcE

2827 10310673.18.2 E

JE 10102.4