LLG
Paris–Abu Dhabi
Advanced Math and Science Pilot Class
Academic year 2014-2015
Assignments - Physics
Assignments: Newton’s law
Data:
I.
g = 9.81 kg.m.s-2 (Earth gravity)
MCQ
1. When starting up, a scooter goes from 0 to 36 km/h in 10s. Its acceleration is:
a) 3.6 m/s2
b) 3.6 km/h2
c) 1 m/s2
Vx(m/s)
2. The graph below shows the evolution of the Ox
coordinate (Vx ) of the velocity vector of a
mobile point moving along the Ox axis. The
graph corresponding to:
- A uniform motion is : a) 1 b) 2 c) 3
- An accelarated motion is: a) 1 b) 2 c) 3
- A uniformely accelerated motion is: a) 1 b) 2 c) 3
2
3
3. The second law of Newton states that for a
material point whose mass m is constant and
whose velocity is 𝑣⃗ :
⃗⃗
𝑑𝑣
⃗⃗⃗⃗
a) 𝐹⃗ = 0
b) 𝐹⃗ = 𝑚 × 𝑑𝑡
1
Time (s)
c) 𝐹⃗ = 𝑚 × 𝑣⃗
4. For the third law of Newton to be applied between two bodies in interaction, the two bodies must:
a) be in contact
b) have the same mass
c) there is no specific condition.
II.
Recognizing some specific motions
1. The magnitude of the velocity of a point in motion along a straight line has been recorded at regular
time intervals, the results are given in the table:
T(s)
0
10
20
30
40
V (m/s)
2
4
8
16
32
Is the motion uniformly accelerated?
2. The following graph shows the trajectory of a point launched from O with a initial velocity ⃗⃗⃗⃗
𝑉0 (free
fall condition).
●
B
●
C
●
A
Z (m)
●
O
●
D
X (m)
For the five
positions O to D,
draw the velocity
and acceleration
vectors. No scale
specified but the
size of the vectors
drawn must take
into account the
change (or not) in
their values.
LLG
Paris–Abu Dhabi
Advanced Math and Science Pilot Class
Academic year 2014-2015
Assignments - Physics
3. A point M is launched with a 20 m/s speed. It slows down and comes to a halt after 10s. During this
phase, we assume that its motion is uniformly accelerated.
a) Calculate the value of its constant acceleration.
b) On a diagram, show velocity and acceleration vector at two different moments t1 and t2.
4. Car test
The motion of a car has been recorded when it starts up along a straight line. After analyzing the motion
thanks to a software, the following results are collected:
t(s)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
V(m/s) 0
2
4
6
8
9.5
10.5 11.3 11.6 11.8 11.9 12.0 12.0 12.0
a) Plot the V-t graph
b) The graph can be divided into three parts, identify them on the graph.
c) What is the value of the acceleration during the first part? Describe the motion.
d) What is the value of the acceleration during the last part? Describe the motion.
e) What is the value of the acceleration at time t = 7s.
5. Vertical launch
A small ball is launched vertically at time t = 0s. The coordinate of its displacement along an Oz axis, oriented
g.t2
upward is : z(t) = − 2 + V0z t + z0 with V0z = 6.0 m/s and z0 = 1.2 m
a) What is the location of the ball at time t = 0s?
b) Find the equation of the coordinate of the velocity on the Oz axis (VZ). What is its value at time t= 0s.
c) Is the ball launched upward or downward?
d) At what time does VZ = 0 ? What is then the location of the ball?
e) Find the equation of the coordinate of the acceleration on the Oz axis (aZ). Describe the motion of the
ball.
III.
Free fall
A. Free fall without initial speed.
1. Free fall of a marble.
A marble falls in free fall from a height h = 2.0 m.
The vertical axis is directed downwards, the origin is the departure position of the marble.
a) Find the differential equation of the motion of its center of gravity.
b) Deduce the kinematic equations.
c) How long does it take for the marble to reach the ground?
d) Represent the speed of the marble in terms of time. Give its value at the end of the fall.
2. On the moon.
Astronaut David Scott drops an object on the moon.
a) Is the object in free fall?
b) Give the differential equation of the motion.
c) Deduce the kinematics equations.
d) Calculate the duration of the fall and the speed gained by the object after a fall of 1.5m height.
Data: the gravitational field on the moon is: gL = 1.6N.kg-1.
3. Depth of a well.
a) Fatma drops a stone into a well. The duration of the fall is 2.0s. What is the depth of the well?
b) The same experiment done in another well gives duration of 4.0s for the fall.
Give the depth on this well.
LLG
Paris–Abu Dhabi
Advanced Math and Science Pilot Class
Academic year 2014-2015
Assignments - Physics
B. Free fall with an initial speed.
1. Juggler.
A juggler launches a ball up vertically with an initial speed V0 = 5.0 m.s-1.
The origin of the dates is the moment of launch. The Oz axis is oriented upwards, the origin O corresponds to
the position of the ball when it is thrown.
a. Find the kinematic equations of the ball assimilated to a point.
b. What is the maximum altitude reached by the ball?
c. How long before the juggler has to catch back the ball in O? Then what is its speed?
d. What must be the initial speed so that the maximum height reached would be 2.0 m?
2. Range and peak height.
A projectile is thrown from a point O located on the ground with an initial speed ⃗⃗⃗⃗⃗⃗
𝑉0 and an angle α with the
horizontal.
z (m)
x (m)
We consider that it is a free fall.
a. Determine the literal expressions of the coordinates of the point S located at the peak on the
trajectory.
b. Determine also the coordinates of the point P located on the horizontal axis on the trajectory.
c. The peak height is the maximum height reached by the projectile. Calculate its value.
d. The range is the distance between points O and P. Calculate its value.
e. The value of the initial speed being constant, for which value of α, will the range be maximal?
Data: V0 = 10 m.s-1; α = 50˚.
3. Throw of two projectiles.
⃗⃗) so that
Two projectiles are thrown simultaneously. Their motion is studied in a coordinate system (O,i⃗, ⃗j, k
⃗⃗) is directed upwards and the point O is on the ground.
the vertical axis (O, k
- The projectile 1 is thrown from point A with coordinates (0; 0; 2.0 m) with an initial speed ⃗⃗⃗⃗⃗⃗⃗⃗
𝑉01 horizontal
and its value is 10 m.s-1.
- The projectile 2 is thrown from point B with coordinates (1.0 m; 0; 2.0 m) with an initial speed ⃗⃗⃗⃗⃗⃗⃗⃗
𝑉02
vertical upwards and its value is 10 m.s-1.
We consider this is a free fall:
a. Establish the kinematic equations of the motion of the 2 projectiles.
b. Compare the durations of the fall and the speed of the two projectiles once they reach the ground.
LLG
Paris–Abu Dhabi
Advanced Math and Science Pilot Class
Academic year 2014-2015
Assignments - Physics
4. Tennis ball.
A player launches a ball vertically and hits it with his racket when its center of gravity is located at a height H
= 2.50 m from the ground. He then provides it a horizontal speed of value V0 = 20 m.s-1.
We assume that the force of friction due to air is neglected.
net
a. Does the ball go over the net located at D = 12.0 m from the launch point? The height of the net at this
point is h = 90.0 cm.
b. The player whishes the ball went 10.0 cm over the net. Which value should he provide to the initial
speed always considered horizontal? How far from the net does the ball touch the ground?
5. Range of fountain.
A device for watering located at the ground level throws a fountain with an initial
speed
V0 = 10 m.s-1 with an angle α = 30˚ with the horizontal. We study the motion of
one
of the drops constituting the fountain.
a. Establish the kinematic equations and the equation of the trajectory of this drop of water.
b. What height h does the fountain rise?
c. The range D is the distance between the start point and the point reached by the drop in the same
horizontal plane. What is the range of this fountain?
d. The value of the initial speed being constant, from which angle α should we incline the device to
water as far as possible? Calculate the value of the range in these conditions.
6. Volley ball service.
Let’s study the center of gravity of a ball in volley ball play. We neglect the air friction.
The player hits the ball located in A and gives it an initial speed ⃗⃗⃗⃗⃗⃗
𝑉0 of value 15 m.s-1 with an angle α = 20˚
with the horizontal.
Point A is at a height H = 3.0 m from the ground; the net is located at D = 12 m from the player, the height of
the net is h = 2.4 m.
net
a. Show that the service is successful, which means that the ball is going over the net and touch the
ground on the other side between the net and the line located at
D’ = 9 m from the net.
b. A player located at 2.0 m from the net wants to intercept the ball. At which height H’ should he put
his hand in the plan of the trajectory of the ball?
Data: mass of the ball: m = 280g, radius of the ball: r = 11 cm.
LLG
Paris–Abu Dhabi
Advanced Math and Science Pilot Class
Academic year 2014-2015
Assignments - Physics
7. About a “thought experiment” of Galilee.
Let’s study the fall of a ball launched from the mast of a ship. Forces exerted by air on the ball are neglected
compared to its weight.
At the date t0 = 0s, the sailor drops the ball from a point O located 14 m above the deck of the ship.
When the ship does not move compared to the dock, the ball falls vertically to the foot of the mast. The
purpose of this exercise is to answer the following question:
“What is happening when the ship is moving?”
We consider the case when the ship has a constant speed in a straight line compared to Earth with a horizontal
speed ⃗⃗⃗⃗⃗⃗
𝑉0 equals 5.0 m.s-1.
a. Study of the trajectory taking the ship as reference.
i. Give the kinematic equations of the motion of the center of gravity of the ball using the coordinate
system drawn on the picture.
ii. Deduce the duration of the fall and the abscissa of the fall point on the ship’s deck.
b. Study of the trajectory in the Earth frame of reference.
i. Give the initial conditions in this frame of reference.
ii. Establish the same kinetic equations of the motion of the center of gravity of the ball and the
equation of the trajectory.
iii. Does the ball fall at the foot of the mast? Justify.
c. Galilee and the relativity of motion.
We will quote a passage from Galileo's Dialogue Concerning the Two Chief World Systems (Dialogo sopra i
due massimi sistemi del mondo). In this text, he uses the example of the ship to demonstrate that the Earth is
moving. His opponents are convinced that Earth does not move.
SALVIATI: You say, then, that since when the ship stands still the rock falls to the foot of the mast, and when
the ship is in motion it falls apart from there, then conversely, from the falling of the rock at the foot it is
inferred that the ship stands still, and from its falling away it may be deduced that the ship is moving. And
since what happens on the ship must likewise happen on the land, from the falling of the rock at the foot of the
tower one necessarily infers the immobility of the terrestrial globe.
SIMPLICIO: That is exactly it, briefly stated, which makes it easy to understand.
SALVIATI: Now tell me: If the stone dropped from the top of the mast when the ship was sailing rapidly fell in
exactly the same place on the ship to which it fell when the ship was standing still, what use could you make of
this falling with regard to determining whether the vessel stood still or moved?
SIMPLICIO: Absolutely none; just as by the beating of the pulse, for instance, you cannot know whether a
person is asleep or awake; the pulse beats in the same manner in sleeping as in waking.
SALVIATI: Very good. Now, have you ever made this experiment on the ship?
SIMPLICIO: I have never made it, but I certainly believe that the authorities who adduced it had carefully
observed it. …
SALVIATI: This is exactly what happened and on the land it’s the same so how can we deduce whether the
globe is moving or not?
i.
ii.
iii.
Which character defends the ideas of Galileo?
With intense fogger weather could we know if the ship is going in a straight line with a constant speed
or if it is at rest?
How does this experiment of Galilee allow us to refute the argument of opponents of Galilee as
immobilize or movement of the earth?
LLG
Paris–Abu Dhabi
Advanced Math and Science Pilot Class
Academic year 2014-2015
Assignments - Physics
8. Mohamed and his pistol arrows.
Mohamed decides to use his knowledge in mechanics to study an arrow shot by its pistol.
Neglecting the air friction and taking g = 10 m.s-2 for the gravity, he considers the arrow
A as a punctual object of mass m = 50 g and with an initial speed⃗⃗⃗⃗⃗⃗
𝑉0 .
The arrow is shot from a point M, at the distance d above the ground, with an initial
speed⃗⃗⃗⃗⃗⃗
𝑉0 making an angle α above horizontal.
a. Theoretical study of the motion of an arrow.
i. In the coordinate system (O, ⃗i, ⃗j), establish the literal kinematic equations of the
motion of an arrow after its launch. Deduce the equation of the trajectory and its nature. The initial state is the
one when the arrow is at the point M.
ii. Youssef is standing on the ground at 15 m right from the point O. He is 1.20 m high. Can he be
touched by the arrow if the arrow is shot from M1 so that d1 = 1.50 m, with an initial speed of 10 m.s-1, the
angle α being equal to 45˚?
b. Experimental determination of Vo.
To determine experimentally Vo, Mohamed did two tests:
i. Vertical shooting.
Gun barrel is vertical, its extremity is located at point M2 so that
d2 = 1.70m.
Mohamed shoots upwards and notices that the arrow touches the
ground 2.20s after
its departure from M2.
Calculate the value Vo of its initial speed by using the kinematic
equations of the
first part.
ii. Horizontal shooting.
To shoot horizontally, Mohamed lowers the gun. Its barrel is now horizontal; its
extremity is located at point M3 so that d3 = 1.20 m. Mohamed shoots and notices that
the arrow touches the ground at point B located on the same horizontality as O at a
distance equals
4.9 m.
Calculate the value of the initial speed by using the kinematic equations of the first
part.