Projectile motion can be thought of as the combination of horizontal and vertical motion. The horizontal motion is unaffected by gravity and follows the equation x=u*cosθ*t, while the vertical motion is affected by gravity and follows the equation y=u*sinθ*t - 0.5*g*t^2. The path of a projectile forms a parabolic curve. Factors like the launch angle and initial velocity determine the time of flight, maximum height, and horizontal range of the projectile. Projectile motion has applications in various sports to calculate the optimal angle for maximum distance.
This PPT covers projectile motion of an object in a very systematic and lucid manner. I hope this PPT will be helpful for instructors as well as students.
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This presentation covers vertical motion under gravity, effect of air resistance on free fall & graphs of free fall. I hope this PPT will be helpful for instructors as well as students.
Describes displacement, velocity, acceleration as vectors and distance and speed as scalars, Show all needed equations and their use.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
This PPT covers projectile motion of an object in a very systematic and lucid manner. I hope this PPT will be helpful for instructors as well as students.
Almost all the things in our daily life are possible because some sort of force. Presence and even absence of force play important role in different situations. This presentation is about 'Forces'. It describes all the types of forces with appropriate and familiar examples.
This presentation covers vertical motion under gravity, effect of air resistance on free fall & graphs of free fall. I hope this PPT will be helpful for instructors as well as students.
Describes displacement, velocity, acceleration as vectors and distance and speed as scalars, Show all needed equations and their use.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
Curvilinear motion occurs when a particle moves along a curved path.
Since this path is often described in three dimensions, vector analysis will
be used to formulate the particle's position, velocity, and acceleration
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For more information, visit. www.vavaclasses.com
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Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
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3. The motion of a projectile may be thought of as
the result of horizontal and vertical
components.
Both the components act independently
What is Projectile Motion?
5. Equation of path of projectile
Suppose at any time t, the object is at point P (x, y).
For motion along horizontal direction, the acceleration ax is zero.
The position of the object at any time t is given by,
Here, x0 = 0, ux = u cos θ, ax = 0
[ Velocity of an object in the horizontal direction is constant]
Putting these values in equation (i),
⇒ x = ut cos θ
For motion along vertical direction, the acceleration ay is −g.
The position of the object at any time t along the vertical direction is
given by,
Here,
6. ∴
Putting the value of t from equation (ii),
⇒
This is an equation of a parabola. Hence, the path of
the projectile is a parabola
7. Time of flight
Total time for which the object is in flight
It is denoted by T.
Total time of flight = Time of ascent + Time of descent
∴ T = t + t = 2t [Time of ascent = Time of descent = t]
At the highest point H, the vertical component of velocity
becomes zero. For vertical motion of the object (from 0 to H),
⇒
∴
8. Maximum height
•Maximum height ‘h’ reached by the projectile
For vertical upward motion from 0 to H,
Using the relation
we obtain
⇒
That is,
9. Horizontal Range
•Horizontal distance covered by the object between its point of projection and the point of hitting the
ground. It is denoted by R.
‘R’ is the distance travelled during time of flight T.
⇒
⇒
For the maximum horizontal range,
sin 2θ = 1 = sin 90°
⇒ 2θ = 90°
⇒ θ = 45°
∴ Maximum horizontal range (Rm
) is
12. x
y
0
Frame of reference:
h
v0
Equations of motion:
X
Uniform m.
Y
Accel. m.
ACCL. ax = 0 ay = g = -9.81
m/s2
VELC. vx = v0 vy = g t
DSPL. x = v0 t y = h + ½ g t2
g
13. Important points :
In case of projectile motion , the horizontal
component of a velocity (u cos θ) , acceleration
(g) and mechanical energy remains constant
while, speed ,velocity , vertical component of
velocity (v sin θ) ,momentum ,kinetic energy
and potential energy all change.
Velocity and K.E are maximum at the point of
projection while minimum (but not zero) at the
highest point.
14. Vector diagrams for
projectile motion
TIME HORIZONTAL VELOCITY VERTICAL VELOCITY
0 s 73.1 m/s, right 19.6 m/s, up
1 s 73.1 m/s, right 9.8 m/s, up
2 s 73.1 m/s, right 0 m/s
3 s 73.1 m/s, right 9.8 m/s, down
4 s 73.1 m/s, right 19.6 m/s, down
5 s 73.1 m/s, right 29.4 m/s, down
6 s 73.1 m/s, right 39.2 m/s, down
7 s 73.1 m/s, right 49.0 m/s, down
15. What two factors would affect projectile motion?
◦ Angle
◦ Initial velocity
Initial Velocity
Angle
16. Evaluating various info
Determination of the Time of Flight
Determination of Horizontal Displacement
x = vix • t
Determination of the Peak Height
y = viy • t + 0.5 • g • t2