Module 2- KINEMATICS
Marlon Flores Sacedon
Visayas State University
Department of Physics
Visca, Baybay City, Leyte
Kinematics of Motion in One-Dimension
QUESTIONS:
o How long will it take to travel?
o What is the distance traveled?
o How fast does it move?
o Etc.
Kinematics of Motion in One-Dimension
Topics
o Introduction to kinematics
o Uniformly Accelerated Bodies (UAB)
o Freely-Falling Bodies (FFB)
o Projectile Motion
Kinematics of Motion in One-Dimension
o Introduction to kinematics
What is Kinematics?
o A branch of classical mechanics that
deals with the study of the motion of
particle without considering the force
that causes them to move
o A purely descriptive study of
motion
o This will answers the question,
how object moves?
Kinematics of Motion in One-Dimension
o Introduction to kinematics
Why study Kinematics?
Kinematics is a significant field of
mechanics that helps us understand how
objects move. By understanding the
concepts involved in kinematics, we can
predict an object's future motion and
solve related problems.
Kinematics of Motion in One-Dimension
o Introduction to kinematics
How to describe motion?
❑ Time (𝑡)
❑ Position (𝑥)
❑ Velocity (𝑣)
❑ Acceleration (𝑎)
❑ 𝑥𝑡, 𝑣𝑡, and 𝑎𝑡 Graphs
Kinematics of Motion in One-Dimension
o Introduction to kinematics
Definition of terms
❑ Displacement is a change in position
∆𝑥 = 𝑥𝑓 − 𝑥𝑖
❑ Velocity is a change in position per time interval
∆𝑥
𝑣𝑎𝑣𝑒 =
∆𝑡
𝑑𝑥
𝑣𝑖𝑛𝑠 =
𝑑𝑡
❑ Acceleration is a change in velocity per time interval
∆𝑣
𝑎𝑎𝑣𝑒 =
∆𝑡
𝑑𝑣
𝑎𝑖𝑛𝑠 =
𝑑𝑡
Kinematics of Motion in One-Dimension
o Introduction to kinematics
Examples of 𝑥𝑡, & 𝑣𝑡- Graphs
𝑥𝑡- Graphs
v𝑡- Graphs
Kinematics of Motion in One-Dimension o Uniformly Accelerated Bodies
What is Uniformly Accelerated Bodies?
Uniformly Accelerated Bodies (UAB) is
a study of motion of particle at
constant acceleration.
𝑑𝑣
𝑎𝑖𝑛𝑠 =
𝑑𝑡
𝑑𝑣
𝑎=
𝑑𝑡
Acceleration is constant
Kinematics of Motion in One-Dimension o Uniformly Accelerated Bodies
Six Physical Quantities of UAB
1. Time interval, 𝑡 ------------ (𝑠)
2. Displacements, s ---------- (𝑚)
3. Average velocity, 𝑣𝑎𝑣𝑒 ---- (𝑚/𝑠)
4. Initial velocity, 𝑣𝑜 ---------- (𝑚/𝑠)
5. Final velocity, 𝑣 ------------ (𝑚/𝑠)
6. Acceleration 𝑎 ------------- (𝑚/𝑠 2 )
Six Formulas of UAB
𝑠
1) 𝑣𝑎𝑣𝑒 =
𝑡
𝑣+𝑣0
2) 𝑣𝑎𝑣𝑒 =
2
𝑣−𝑣0
3) 𝑎 =
𝑡
5) 𝑠 = 𝑣0 𝑡 + 1Τ2 𝑎𝑡 2
4) 𝑣 = 𝑣0 + 𝑎𝑡
6) 𝑣 2 = 𝑣0 2 + 2𝑎𝑠
Kinematics of Motion in One-Dimension o Uniformly Accelerated Bodies
PROBLEMS in UAB
1) Given: 𝑠 = 10 m ; 𝑡 = 3 𝑠 ; 𝑎 = 2 m/s2
Required: 𝑣𝑜 , 𝑣, 𝑣𝑎𝑣𝑒
Kinematics of Motion in One-Dimension o Uniformly Accelerated Bodies
PROBLEMS in UAB
2) Given: 𝑠 = 150 cm ; 𝑡 = 2 𝑠 ; 𝑣𝑜 = 14 cm/s
Required: 𝑎, 𝑣𝑎𝑣𝑒 , 𝑣
Kinematics of Motion in One-Dimension o Uniformly Accelerated Bodies
Quiz #2 (5 pts)
Given: 𝑠 = 50 m ; 𝑎 = 2m/s2; 𝑣𝑜 = 9 m/s
Required: t, 𝑣𝑎𝑣𝑒 , 𝑣
Kinematics of Motion in One-Dimension o Uniformly Accelerated Bodies
Problem: An antelope moving with constant
acceleration covers the distance between two
points 60.0 𝑚 apart in 5.00 𝑠. Its speed as it passes
the second point is 14.0 𝑚/𝑠. What are (a) its
speed at the first point and (b) its acceleration?
Kinematics of Motion in One-Dimension o Uniformly Accelerated Bodies
Applications
PROBLEM: A bird is flying due east. Its distance from a tall
building is given by 𝑥 𝑡 = 28.0𝑚 + 12.4𝑚/𝑠 𝑡 −
(0.045 𝑚/𝑠 3 )𝑡 3 . What is the instantaneous velocity of the
bird when 𝑡 = 8.00 𝑠?
Kinematics of Motion in One-Dimension o Uniformly Accelerated Bodies
Try this: The graph in Figure
shows the velocity of a
motorcycle police officer plotted
as a function of time. (a) Find
the instantaneous acceleration
at 𝑡 = 3 𝑠, 𝑡 = 7 𝑠, and 𝑡 = 11 𝑠.
(b) How far does the officer go in
the first 5 𝑠? The first 9 𝑠? The
first 13 𝑠?
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
▪ Freely falling bodies is an object that is falling
under the sole influence of gravity. Any object
that is being acted upon only by the force of
gravity is said to be in a state of free fall.
▪ Air resistance and other external forces are neglected.
▪ In the absence of air resistance, regardless its size
and shape, the body will fall at the same rate.
▪ Free-fall is a good example of Uniformly Accelerated
Bodies (UAB).
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
Gravitation acceleration
22
+𝑦
𝑔 = −9.81
9.81 𝑚/𝑠
𝑚/𝑠 +𝑦
0
0
−𝑦
−𝑦
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
Six Physical Quantities of FFB
1. Time interval, 𝑡 ------------ (𝑠)
2. Displacements, 𝑦 ---------- (𝑚)
3. Average velocity, 𝑣𝑎𝑣𝑒 ---- (𝑚/𝑠)
4. Initial velocity, 𝑣𝑜𝑦 ---------- (𝑚/𝑠)
5. Final velocity, 𝑣𝑦 ------------ (𝑚/𝑠)
6. Gravitational Acceleration 𝑔 = −9.81 𝑚/𝑠 2
Six Formulas of UAB
𝑠
1) 𝑣𝑎𝑣𝑒 =
𝑡
2) 𝑣𝑎𝑣𝑒 =
3) 𝑔 =
𝑣𝑦 +𝑣0𝑦
2
𝑣𝑦 −𝑣0𝑦
𝑡
4) 𝑣𝑦 = 𝑣0𝑦 + 𝑔𝑡
5) 𝑦 = 𝑣0𝑦 𝑡 + 1Τ2 𝑔𝑡 2
6) 𝑣𝑦 2 = 𝑣0𝑦 2 + 2𝑔𝑠
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
PROBLEM: A small rock is thrown vertically upward with a speed of
22.0 m/s from the edge of the roof of a 30.0-m-tall building. The rock
doesn’t hit the building on its way back down and lands on the street
below. Ignore air resistance. (a) What is the speed of the rock just
before it hits the street? (b) How much time elapses from when the
rock is thrown until it hits the street?
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
PROBLEM: You throw a small rock straight up from the edge
of a highway bridge that crosses a river. The rock passes you on
its way down, 6.00 s after it was thrown. What is the speed of
the rock just before it reaches the water 28.0 m below the point
where the rock left your hand? Ignore air resistance.
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
PROBLEM: You throw a glob of putty straight up toward
the ceiling, which is 3.60 m above the point where the
putty leaves your hand. The initial speed of the putty as it
leaves your hand is 9.50 m/s. (a) What is the speed of the
putty just before it strikes the ceiling? (b) How much time
from when it leaves your hand does it take the putty to
reach the ceiling?
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
1) You throw a small rock straight up from the edge of a highway bridge that crosses
a river. The rock passes you on its way down, 6.00 s after it was thrown. What is the
speed of the rock just before it reaches the water 28.0 m below the point where the
rock left your hand? Ignore air resistance.
2) A flowerpot falls off a windowsill and passes the window of the story below. Ignore
air resistance. It takes the pot 0.380 s to pass from the top to the bottom of this
window, which is 1.90 m high. How far is the top of the window below the windowsill
from which the flowerpot fell?
3) A small rock is thrown vertically upward with a speed of 22.0 m/s from the edge of the roof
of a 30.0-m-tall building. The rock doesn’t hit the building on its way back down and lands on
the street below. Ignore air resistance. (a) What is the speed of the rock just before it hits the
street? (b) How much time elapses from when the rock is thrown until it hits the street?
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
Problems
1) (a) If a flea can jump straight up to a height of 0.440
m, what is its initial speed as it leaves the ground? (b)
How long is it in the air?
2) A juggler throws a bowling pin straight up with an
initial speed of 8.20 m/s. How much time elapses until
the bowling pin returns to the juggler’s hand?
Kinematics of Motion in One-Dimension o Freely-falling Bodies (FFB)
Freely-falling Bodies (FFB)
ASSIGNMENT
1) (a) If a flea can jump straight up to a height of 0.440 m, what is its initial speed as
it leaves the ground? (b) How long is it in the air?
ANSWER:
2) A 15-kg rock is dropped from rest on the earth and reaches the ground in 1.75 s.
When it is dropped from the same height on Saturn’s satellite Enceladus, the rock
reaches the ground in 18.6 s. What is the acceleration due to gravity on Enceladus?
ANSWER:
3) A juggler throws a bowling pin straight up with an initial speed of 8.20 m/s. How much time
elapses until the bowling pin returns to the juggler’s hand?
ANSWER:
Projectile Motion
The Idealized Model
Kinematics on 2D or 3D Motion of Point Particle
QUESTION # 1: How many possible angle/s of
projection in order to shot the ball into the
cylinder’s opening?
QUESTION #2: What are the magnitudes of
these angles?
12 𝑚/𝑠
𝜃2 = ?
2𝑚
𝜃1 = ?
7𝑚
CANNON
TARGET
OPEN CYLINDER
To answer… Study Projectile Motion
Learning Outcomes
Derive the formulas for Idealized model of
Projectile Motion and apply to real life problems.
6. 𝑥 = 𝑣𝑜𝑥 ∙ 𝑡
1. 𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
1 2
7. 𝑦 = 𝑣𝑜𝑦 ∙ 𝑡 − 2𝑔𝑡
2. 𝑣 = 𝑣 sin 𝜃
𝑜𝑦
𝑜
3. 𝑣𝑥 = 𝑣𝑜𝑥
4. 𝑣𝑦 = 𝑣𝑜𝑦 − 𝑔𝑡
5. 𝑣 =
𝑣𝑥 2 + 𝑣𝑦 2
8.
𝑥
𝑡=
𝑣𝑜 cos 𝜃
𝑔𝑥 2
𝑦 = 𝑥 tan 𝜃 − 2𝑣 2cos2𝜃
𝑜
Learning Outcomes
Derive the formulas for Idealized model of
Projectile Motion and apply to real life problems.
Tricky problem
Recall:
❖ Kinematics of Motion
❖ Describing motion of particle/s
❖ Motion
❖ Change in positions
❖ Position, time, velocity, & acceleration
❖ Kinematics in One-Dimensional Motion
o Uniformly accelerated bodies
o Freely Falling Bodies
Recall:
❖ Some Trigonometric Identities
o
1
cos 𝜃 = 2
sec 𝜃
2
2
2
o sec 𝜃 = 1 + tan 𝜃
1
cos2 𝜃
o
sec 2 𝜃 =
o
sin𝜃
tan𝜃 =
cos𝜃
What is Projectile Motion?
Trajectory
+𝑦
𝑥, 𝑦, 𝑡
𝑣𝑜
𝜃
𝑦
+𝑥
𝑥
Projectile Motion
Time of Flight
Initial Velocity of projected body
+𝑦
Angle of projection
𝑡
Position y
𝑥, 𝑦, 𝑡
𝑣𝑜
Position 𝑥
𝜃
+𝑥
𝑥
𝑦
Projectile Motion: The Idealized Model
ASSUMPTIONS:
❖ Air resistance & other effects are negligible
+𝑦
𝑡
𝑥, 𝑦, 𝑡
𝑣𝑜
𝑦
𝜃
+𝑥
𝑥
▪ Along 𝑥-axis: Uniform motion 𝒂𝒙 = 𝟎
▪ Along y-axis: Free-fall 𝒂𝒚 = 𝒈 = −𝟗. 𝟖𝟏 m/s2
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
𝑣𝑦 = 0
𝑣𝑦
+𝑦
𝑣𝑜𝑦
𝑣
𝑣𝑥
𝑣𝑜
𝜃
𝑣𝑜𝑥
𝑣
𝑣𝑥
𝑣𝑥
𝑥, 𝑦, 𝑡
𝑔 = −9.81
𝑣𝑦
𝑣
𝑦
+𝑥
𝑥
▪ Along 𝑥-axis: Uniform motion 𝒂𝒙 = 𝟎
▪ Along y-axis: Free-fall 𝒂𝒚 = 𝒈 = −𝟗. 𝟖𝟏 m/s2
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
+𝑦
Let’s derive the formulas
𝑡
𝑥, 𝑦, 𝑡
𝑣𝑜𝑦
𝑣𝑜
𝑣𝑜
𝑣𝑜𝑦
𝑣𝑜𝑥
𝑔 = −9.81
𝑦
𝜃
+𝑥
𝑥
𝜃
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
+𝑦
𝑡
𝑥, 𝑦, 𝑡
𝑣𝑜𝑦
𝑣𝑜
𝑔 = −9.81
𝑦
𝜃
𝑣𝑜𝑥
𝑣𝑜𝑥
+𝑥 cos 𝜃 =
𝑣𝑜
𝑥
1. 𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
𝑣𝑜
𝜃
𝑣𝑜𝑥
𝑣𝑜𝑦
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
+𝑦
𝑡
𝑥, 𝑦, 𝑡
𝑣𝑜𝑦
𝑣𝑜
𝑔 = −9.81
𝑦
𝜃
𝑣𝑜𝑥
1. 𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
2. 𝑣𝑜𝑦 = 𝑣𝑜 sin 𝜃
𝑣𝑜𝑦
+𝑥 sin 𝜃 =
𝑣𝑜
𝑥
𝑣𝑜𝑦 = 𝑣𝑜 sin 𝜃
𝑣𝑜
𝜃
𝑣𝑜𝑥
𝑣𝑜𝑦
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
𝑣𝑦
+𝑦
𝑣
𝑣𝑥
𝑣𝑜𝑦
𝑣𝑦
𝑣𝑜
𝑡
𝑥, 𝑦, 𝑡
𝑔 = −9.81
𝑦
𝜃
𝑣𝑣𝑥𝑜𝑥
1. 𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
2. 𝑣𝑜𝑦 = 𝑣𝑜 sin 𝜃
+𝑥
𝑥
𝑑𝑣
𝑎=
𝑑𝑡
𝑡𝑣
න: 𝑑𝑣 =
න
= 𝑎 𝑑𝑡
0
𝑣𝑜
𝑣 − 𝑣𝑜 = 𝑎𝑡
𝑣 = 𝑣𝑜 + 𝑎𝑡
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
𝑣𝑦
+𝑦
𝑣
𝑣𝑥
𝑣𝑜𝑦
𝑣𝑜
𝑡
𝑥, 𝑦, 𝑡
𝑔 = −9.81
𝑦
𝜃
𝑣𝑜𝑥
1. 𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
2. 𝑣𝑜𝑦 = 𝑣𝑜 sin 𝜃
+𝑥
𝑥
3. 𝑣𝑥 = 𝑣𝑜𝑥
𝑣𝑥 = 𝑣𝑜𝑥 + 0 𝑡
𝑣𝑥 = 𝑣𝑜𝑥
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
𝑣𝑦
+𝑦
𝑣
𝑣𝑥
𝑣𝑜𝑦
𝑣𝑜
𝑡
𝑥, 𝑦, 𝑡
𝑔 = −9.81
𝑦
𝜃
𝑣𝑜𝑥
1. 𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
2. 𝑣𝑜𝑦 = 𝑣𝑜 sin 𝜃
+𝑥
𝑥
3. 𝑣𝑥 = 𝑣𝑜𝑥
4. 𝑣𝑦 = 𝑣𝑜𝑦 − 𝑔𝑡
𝑣𝑦 = 𝑣𝑜𝑦 + −𝑔 𝑡
𝑣𝑦 = 𝑣𝑜𝑦 − 𝑔𝑡
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
𝑣𝑦
+𝑦
𝑣
𝑣𝑥
𝑣𝑜𝑦
𝑣𝑜
𝑡
𝑥, 𝑦, 𝑡
𝑔 = −9.81
𝑦
𝜃
𝑣𝑜𝑥
1. 𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
2. 𝑣𝑜𝑦 = 𝑣𝑜 sin 𝜃
+𝑥
𝑥
5. 𝑣 =
3. 𝑣𝑥 = 𝑣𝑜𝑥
4. 𝑣𝑦 = 𝑣𝑜𝑦 − 𝑔𝑡
𝑣𝑥 2 + 𝑣𝑦 2
𝑟𝑡
𝑣 = 𝑣𝑜 + 𝑎𝑡
𝑑𝑟
= 𝑣𝑜𝑜 + 𝑎𝑡
𝑎𝑡
𝑑𝑡
න:
න 𝑑𝑟 = 𝑣𝑜 + 𝑎𝑡 𝑑𝑡
0
1 2
𝑟 = 𝑣𝑜 𝑡 + 2𝑎𝑡
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
𝑔 = −9.81
Trajectory is a parabolic curve
+𝑦
𝑡
𝑥, 𝑦, 𝑡
𝑣𝑜𝑦
𝑣𝑜
𝑦
𝜃
𝑣𝑜𝑥
6. 𝑥 = 𝑣𝑜𝑥 ∙ 𝑡
1 2
7. 𝑦 = 𝑣𝑜𝑦 ∙ 𝑡 − 2𝑔𝑡
𝑥 = 𝑣𝑜𝑥 ∙ 𝑡 1 2
+𝑥
𝑦 = 𝑣𝑜𝑦 ∙ 𝑡 − 2𝑔𝑡
𝑥
𝑥
8. 𝑡 =
𝑥 = 𝑣𝑜 cos 𝜃 ∙ 𝑡
𝑣𝑜 cos 𝜃
𝑦 = 𝑣𝑜 sin 𝜃 ∙ 𝑡 − 12𝑔𝑡 2
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
𝑔 = −9.81
+𝑦
𝑡
𝑥, 𝑦, 𝑡
𝑣𝑜𝑦
𝑣𝑜
𝑦
𝜃
𝑣𝑜𝑥
+𝑥
𝑥
𝑥
8. 𝑡 =
𝑥 = 𝑣𝑜 cos 𝜃 ∙ 𝑡
𝑣𝑜 cos 𝜃
1 2
𝑦 = 𝑣𝑜 sin 𝜃 ∙ 𝑡 − 2𝑔𝑡
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
𝑔 = −9.81
+𝑦
𝑡
𝑥, 𝑦, 𝑡
𝑣𝑜𝑦
𝑣𝑜
𝑦
𝜃
𝑣𝑜𝑥
+𝑥
𝑥
8. 𝑡 =
𝑥
𝑣𝑜 cos 𝜃
𝑥 = 𝑣𝑜 cos 𝜃 ∙ 𝑡
𝑥
𝑔𝑥12
𝑥
𝑦
=
𝑥
tan
𝜃
−
𝑦 = 𝑣𝑜 sin 𝜃 ∙
−
𝑔(
)2
2
2
2 𝑣𝜃 cos 𝜃
2𝑣 cos
𝑣𝑜 cos 𝜃𝑜
𝑜
𝑔𝑥 2
𝑦 = 𝑥 tan 𝜃 − 2𝑣 2cos2𝜃
𝑜
Time independent formula
Projectile Motion: The Idealized Model
ASSUMPTIONS: Air resistance & other effects are negligible
FORMULAS
1. 𝑣𝑜𝑥 = 𝑣𝑜 cos 𝜃
2. 𝑣𝑜𝑦 = 𝑣𝑜 sin 𝜃
3. 𝑣𝑥 = 𝑣𝑜𝑥
4. 𝑣𝑦 = 𝑣𝑜𝑦 − 𝑔𝑡
5. 𝑣 =
𝑣𝑥 2 + 𝑣𝑦 2
6. 𝑥 = 𝑣𝑜𝑥 ∙ 𝑡
7. 𝑦 = 𝑣𝑜𝑦 ∙ 𝑡 −
1 2
𝑔𝑡
2
8.
𝑥
𝑡=
𝑣𝑜 cos 𝜃
𝑔𝑥 2
𝑦 = 𝑥 tan 𝜃 − 2𝑣 2cos2𝜃
𝑜
Time independent formula
APPLICATION 1: An object will be projected at an
angle of 65o above the horizontal surface at a speed
of 13 m/s. Ignoring air resistance, (a) what will be
the horizontal displacement of the particle 1.5
second after being released? (b) What will be
velocity during 1.5 seconds of flight? (c) What will
be the maximum height of the particle above the
starting point?
APPLICATION 2:
o
THE TRICKY PROBLEM
o
1
cos 𝜃 = 2
sec 𝜃
2
2
sec 𝜃 = 1 + tan2 𝜃
2 = 7 tan 𝜃 − 1.669 sec 2 𝜃
2 = 7 tan 𝜃 − 1.669(1 + tan2 𝜃)
3.669 = 7 tan 𝜃 − 1.669tan2 𝜃
Let: 𝒛 = 𝐭𝐚𝐧 𝜽
9.81(7)2
2 = 7 tan 𝜃 − 2(12)2 cos2 𝜃
1.669
2 = 7 tan 𝜃 − cos2 𝜃
3.669 = 7𝑧 − 1.669z 2
1.669z 2 − 7𝑧 + 3.669 = 0
z 2 − 4.19𝑧 + 2.2 = 0
TRICKY APPLICATION PROBLEM
o
o
1
cos 𝜃 = 2
sec 𝜃
2
2
sec 𝜃 = 1 + tan2 𝜃
z 2 − 4.19𝑧 + 2.2 = 0
4.19 ± (4.19)2 −4(1)(2.2)
𝑧=
2(1)
Let: 𝒛 = 𝐭𝐚𝐧 𝜽
𝑧1 = 0.615
𝜃1 = 31.59𝑜
𝑧2 = 3.575
𝜃2 = 74.37𝑜
ANSWERS
Time of Flights
8.
𝑥
𝑡=
𝑣𝑜 cos 𝜃
PRACTICE PROBLEM
PROBLEM 1: A boy 12.0 m above the ground in a
tree throws a ball for his dog, who is standing right
below the tree and starts running the instant the ball
is thrown. If the boy throws the ball horizontally at
8.50 m/s, (a) how fast must the dog run to catch the
ball just as it reaches the ground, and (b) how far
from the tree will the dog catch the ball?
PRACTICE PROBLEM
PROBLEM 2: On level ground a shell is fired with an initial
velocity of 40.0 m/s at 60.0° above the horizontal and feels
no appreciable air resistance. (a) Find the horizontal and
vertical components of the shell’s initial velocity. (b) How
long does it take the shell to reach its highest point? (c) Find
its maximum height above the ground. (d) How far from its
firing point does the shell land? (e) At its highest point, find
the horizontal and vertical components of its acceleration
and velocity.
PRACTICE PROBLEM
PROBLEM 3: A man stands on the roof of a 15.0-m-tall
building and throws a rock with a speed of 30.0 m/s at an
angle of 33.0o above the horizontal. Ignore air resistance.
Calculate (a) the maximum height above the roof that the
rock reaches; (b) the speed of the rock just before it strikes
the ground; and (c) the horizontal range from the base of
the building to the point where the rock strikes the ground.
PRACTICE PROBLEM
PROBLEM 4: A rookie quarterback throws a football with an
initial upward velocity component of 12.0 m/s and a
horizontal velocity component of 20.0 m/s. Ignore air
resistance. (a) How much time is required for the football to
reach the highest point of the trajectory? (b) How high is
this point? (c) How much time (after it is thrown) is required
for the football to return to its original level? How does this
compare with the time calculated in part (a)?
PRACTICE PROBLEM
PROBLEM 5: A rock is thrown with a velocity 𝑣𝑜 , at
an angle of 𝑎𝑜 from the horizontal, from the roof of a
building of height ℎ. Ignore air resistance. Calculate
the speed of the rock just before it strikes the
ground, and show that this speed is independent of
𝑎𝑜 .
Assignment
PROBLEM 6: Crickets Chirpy and Milada jump from
the top of a vertical cliff. Chirpy drops downward and
reaches the ground in 2.70 s, while Milada jumps
horizontally with an initial speed of 95.0 cm/s. How
far from the base of the cliff will Milada hit the
ground? Ignore air resistance.
Answer: 2.57 m
QUIZ
APPLICATION 1: A tennis ball rolls off the
edge of a table top 0.75 m above the floor and
strikes the floor at a point 1.40 m horizontally
from the edge of the table. Ignore air
resistance.
a). Find the magnitude of the initial velocity.
b). Find the time of flight.
c). Find the magnitude of the velocity of the
ball just before it strikes the floor
.
Thank you…
Next topic: Newton’s Laws of Motion