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An object projected down at an angle from an elevation

The time of flight is obtained considering motion in vertical direction as :

y = H = u y T + 1 2 g T 2

Rearranging, we get a quadratic equation in “T”,

T 2 + 2 u y g T 2 H 1 g = 0

One of the two values of "T" gives the time of flight in this case. The horizontal range, on the other hand, is given as :

x = R = u x T

Exercises

Two balls of masses " m 1 ” and “ m 2 ”are thrown from a tower in the horizontal direction at speeds " u 1 ” and “ u 2 ” respectively at the same time. Which of the two balls strikes the ground first?

(a) the ball thrown with greater speed

(b) the ball thrown with lesser speed

(c) the ball with greater mass

(d) the balls strike the ground simultaneously

The time to strike the ground is obtained by considering motion in vertical direction.

Here, u y = 0 and y=T (total time of flight)

y = 1 2 g T 2 T = 2 y g

Thus, we see that time of flight is independent of both mass and speed of the projectile in the horizontal direction. The two balls, therefore, strike the ground simultaneously.

Hence, option (d) is correct.

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A body dropped from a height "h" strikes the ground with a velocity 3 m/s. Another body of same mass is projected horizontally from the same height with an initial speed of 4 m/s. The final velocity of the second body (in m/s), when it strikes the earth will be :

a 3 b 4 c 5 d 7

We can use the fact that velocities in vertical direction, attained by two bodies through same displacement, are equal. As such, velocity of the second body , on reaching the ground, would also be 3 m/s. Now, there is no acceleration in horizontal direction. The horizontal component of velocity of the second body, therefore, remains constant.

v x = 4 m / s

and

v y = 3 m / s

The resultant velocity of two component velocities in mutually perpendicular directions is :

v = v x 2 + v y 2 = 4 2 + 3 2 = 5 m / s

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A projectile (A) is dropped from a height and another projectile (B) is projected in horizontal direction with a speed of 5 m/s from the same height. Then the correct statement(s) is(are) :

(a) The projectile (A) reach the ground earlier than projectile (B).

(b) Both projectiles reach the ground with the same speed.

(c) Both projectiles reach the ground simultaneously.

(d) The projectiles reach the ground with different speeds.

First three options pertain to the time of flight. The time of flight depends only the vertical displacement and vertical component of projection velocity. The vertical components of projection in both cases are zero. The time of flight (T) in either case is obtained by considering motion in vertical direction as :

y = u y T + 1 2 a y T 2 H = 0 + 1 2 g T 2 T = 2 H g

Hence, both projectiles reach the ground simultaneously. On the other hand, component of velocity in vertical direction, on reaching the ground is :

v y = u y + a y T = 0 + g T

Putting value of T, we have :

v y = 2 g H

Projectile (A) has no component of velocity in horizontal direction, whereas projectile (B) has finite component of velocity in horizontal direction. As such, velocities of projectiles and hence the speeds of the particles on reaching the ground are different.

Hence, options (c) and (d) are correct.

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Four projectiles, "A","B","C" and "D" are projected from top of a tower with velocities (in m/s) 10 i + 10 j , 10 i - 20 j , -10 i - 10 j and -20 i + 10 j in the coordinate system having point of projection as origin. If "x" and "y" coordinates are in horizontal and vertical directions respectively, then :

(a)Time of flight of C is least.

(b) Time of flight of B is least.

(c) Times of flights of A and D are greatest.

(d) Times of flights of A and D are least.

The time of flight of a projectile solely depends on vertical component of velocity.

Projectile motion

Projectile motion

The projectile "A" is thrown up from an elevated point with vertical component of velocity 10 m/s. It travels to the maximum height ( H 1 ) and the elevation from the ground( H 2 ).

The projectile "B" is thrown down from an elevated point with vertical component of velocity 20 m/s. It travels only the elevation from the ground ( H 2 ).

The projectile "C" is thrown down from an elevated point with vertical component of velocity 10 m/s. It travels only the elevation from the ground ( H 2 ).

The projectile "D" is thrown up from an elevated point with vertical component of velocity 15 m/s. It travels to the maximum height ( H 1 ) and the elevation from the ground( H 2 ).

Clearly, projectile "B" and "C" travel the minimum vertical displacement. As vertical downward component of velocity of "B" is greater than that of "C", the projectile "B" takes the least time.

The projectiles "A" and "D" are projected up with same vertical components of velocities i.e. 10 m/s and take same time to travel to reach the ground. As the projectiles are projected up, they take more time than projectiles projected down.

Hence, options (b) and (c) are correct.

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Acknowledgment

Author wishes to thank Scott Kravitz, Programming Assistant, Connexions for making suggestion to remove syntax error in the module.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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what is Nano technology ?
Bob Reply
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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why?
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biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Kinematics fundamentals. OpenStax CNX. Sep 28, 2008 Download for free at http://cnx.org/content/col10348/1.29
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