2. Many factors have to be taken into account toachieve a successful rocket launch,

maintain a stable orbitand return to Earth

Galileo's analysis of projectile motion

To slow the motion, he used balls rolling sidewards across inclined planes to study projectile motion. (This made the acceleration down the inclined plane slower than the effect of gravity on a thrown ball)

Galileo was the first to accurately describe projectile motion.

He showed that it could be understood by analysing the horizontal and vertical components of motion separately.

The inclined plane experiments involved minimal effect of air resistance so horizontal velocity was constant as he observed the motion across the inclined plane.

He related the time, t, to travel down the inclined plane a distance, d: t2d

Vertical velocity was accelerated by gravity at a constant rate, irrespective of mass.

Based on these observations, Galileo argued that the trajectory of a projectile would be a parabola.

At any time, t, we could add the separate vertical and horizontal components to get resultant velocity.

When an object is projected, we can treat the components separately and apply our equations of motion to each (v = u + at etc.) then add the components vectorially to find the resultant at any point.

Describe Galileo’s analysis of projectile motion

Further analysis would have showed that the horizontal velocity of the arrow was fairly constant (thin arrow not affected much by air resistance) and that both arrows accelerated downwards vertically at the same rate (g).

An example of Galileo's accuracy can be found in an experiment conducted with a previous Physics class.

We had a bow and arrow which one student fired horizontally above flat ground. At (supposedly) exactly the same time, another student dropped an arrow from the same height (holding it horizontally, the same as the one in the bow).

The horizontally fired arrow and the dropped arrow hit the ground at the same time, showing that the vertical velocity was independent of the horizontal velocity.

Describe the trajectory of an object undergoing projectile motion within the Earth’s gravitational field in terms of horizontal and vertical

components

vx= ux+ axtvx

2 = ux2+ 2axx

x = uxt + ½ axt2

vx= ux

vx2 = ux

2

x = uxt

vy= uy+ aytvy

2 = uy2+ 2ayy

y = uyt + ½ ayt2

solve problems and analyse information to calculate the actual velocity of a projectile from

its horizontal and vertical components using:

v = u + at v2 = u2+ 2as

s = ut + ½ at2

(N.B. change to component versions)

Perform a first-hand investigation,gather information and

analyse data to calculate initial and final velocity,

maximum height reached, range,and time of flight of a projectile, for a range of situations by using

simulations, data loggers andcomputer analysis

Projectile motion Experiments

Marble launcher

Data logger for launch velocity

Apple Computer simulation

Video analysisStroboscopic photography

Question 2

In 1642, Galileo devised an experiment to support the heliocentric model

of the universe. An object was dropped from the crows nest of a ship as

illustrated in the diagram below.

The object had a mass of 1.0 kg and the ship was moving relative to the water with a velocity of 5.0 ms-1

west. The crows nest was 15 m high

(a) Determine what the horizontal displacement of the object will be when it reaches the water relative to

the ship.

(b) Determine the magnitude and direction of the velocity of the object relative to the water just before it hit

the surface below.

(c) The object gained kinetic energy as it fell. Determine the amount of kinetic energy the object gained and

explain where the energy came from.

15 m

5 m/s

2 (a) 1 mark

s = 0 m

(b) 3 marks

vv = (u2 + 2ay)0.5

vv = (0 + 2 x 9.8 x 15)0.5

vv = 17.1

vf = (17.12 +52)0.5

vf = 18 ms-1

tan = 17.1/5.0

= 74o

vf = 18 ms-1 at 74o below the horizontal to the west.

(c) 2 marks

KE = ½mv2 - ½mu2

KE = 0.5 x 1 x (17.12 – 52)

KE = 134 J

The object has gravitational potential energy

which is converted into kinetic energy as the

object falls to the water.

15 m

5 m/s

Question 2

A student throws a ball off a cliff with a speed of 30 ms-1 at an angle of 30o.

The ball took 5 s to land at the bottom of the cliff.

(a) Determine the vertical component of the velocity

(b) Determine the height of the cliff.

2 (a) 1 mark

vy = 30 cos 60 = 30 sin 30

vy = 15 ms-1 up

(b) 1 mark

y = uyt + ½ ayt2

y = -15 x 5 + 0.5 x 9.8 x 52

y = 47.5 m

Question 3

The diagram below is the record of a stroboscopic photograph of the motion

of a projectile fired on Earth.

(a) Determine the maximum height reached.

(b) Determine the vertical component of the projectiles initial velocity.

(c) Determine the time of the flight.

(d) Determine the horizontal component of the projectiles initial velocity.

(e) Determine the initial velocity including the direction.

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

horizontal distance (m)

hei

gh

t (m

)

3(a) 1 mark

y = 30 m

(b) 1 mark

uy = (vy2 – 2ay

y)0.5

uy = (0 - -2 x –9.8 x 30)0.5

uy = 24.2 ms-1

(c) 1 mark

t = (vy – uy)/ ay

t = (-24.2 – 24.2)/-9.8

t = 4.95 s

(d) 1 mark

ux = x/t

ux = 34/4.95

ux = 6.87 ms-1

(e) 2 marks

v = (24.22 + 6.872)0.5

v = 25.2 ms-1

= tan-1(24.2/6.87)

= 74o

v = 25.2 ms-1 at 74o above the horizontal.