Lesson Problems on projectile launched at an angle to horizon

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Problems on projectile launched at an angle to horizon

Problem 1

Joe Nedney of the San Francisco 49ers kicked a field goal with an initial velocity of 20 ft/s at an angle of 60°.
(a) How long is the ball in the air? You may assume that the ball lands at the same height as it starts at.
(b) What is its maximum height?
(c) What is the distance traveled by the ball before it hits the ground?

Solution

(a)  Vertical component of the initial velocity is  20*sin(60°) =  = 17.32 ft/s.


    Therefore, the equation for the vertical coordinate h(t) is

        h(t) = -16t^2 + 17.32t


    The equation to find the time of the flight is  h(t) = 0,  or

        -16t^2 + 17.32t = 0,  or

         t*(16t - 17.32) = 0


    We are interested in positive root, ONLY, and this root is


        t =  = 1.0825 seconds.     ANSWER




(b)  To find maximum height, notice that the time moving up is HALF of the total flight time, i.e.,  = 0.54125 seconds.


     To find  , calculate the function h(t) at t = 0.54125:  


          = -16*0.54125^2 + 17.32*0.54125 = 4.687 ft.    ANSWER




(c)  The horizontal component of the speed is  20*cos(60°) =  = 10 ft/s and is considered as a constant during the flight.


     Moving with the horizontal speed of 10 ft/s during 1.0825 seconds, the ball will get the ground at the distance of  

         1.0825*10 = 10.825 feet horizontally from the starting position.      ANSWER

Problem 2

A missile is fired with an initial velocity of 100 m/s at an angle 30° from the horizontal.
(a) After how many seconds will the missile reach its highest point?
(b) What is the maximum height reached by the missile?
(c) Calculate the total horizontal distance travelled by the missile after it hits the ground.

Solution

(a)  Vertical component of the initial velocity is  100*sin(30°) =  = 50 m/s.


    Therefore, the equation for the vertical coordinate h(t) above the ground is

        h(t) = -5t^2 + 50t  meters.


    The equation to find the time of the flight is  h(t) = 0,  or

        -5t^2 + 50t = 0,  or

         5t*(t - 10) = 0


    We are interested in positive root, ONLY, and this root is


        t = 10  seconds.     ANSWER




(b)  To find maximum height, notice that the time moving up is HALF of the total flight time, i.e.,  = 5 seconds.


     To find  , calculate the function h(t) at t = 5:  


          = -5*5^2 + 50*5 = 125 meters.    ANSWER




(c)  The horizontal component of the speed is  100*cos(30°) =  = 86.60 m/s and is considered as a constant during the flight.


     Moving with the horizontal speed of 86.60 m/s during 10 seconds, the missile will get the ground at the distance of  

         86.60*10 = 866 meters horizontally from the starting position.      ANSWER

Problem 3

A projectile is fired at an angle of 30° above the horizontal from the top of a cliff 600 ft high.
The initial speed of the projectile is 2000 ft/s. How far will the projectile move horizontally
before it hits the level ground at the base of the cliff?

Solution

Vertical component of the initial velocity is half of 2000 ft/s, or 1000 ft/s.


Therefore, the equation for the vertical coordinate h(t) above the cliff base level is

    h(t) = -16t^2 + 1000t + 600


The equation to find the time of the flight is  h(t) = 0,  or

    -16t^2 + 1000t + 600 = 0,  or

     4t^2 - 250t - 150 = 0.


Its roots are   =  =  = .



Of these two roots, only positive is interesting for us  t =  = 62.8 seconds  (rounded).



The horizontal component of the speed is   = 1732 ft/s (rouned) and is considered as a constant during the flight.


Moving with the horizontal speed of 1732 ft/s during 62.8 seconds, the projectile will get the ground level at the distance of  

       62.8*1732 = 108769.6 feet from the cliff base.      ANSWER

My other lessons in this site on a projectile thrown/shot/launched vertically up or horizontally are
    - Introductory lesson on a projectile thrown-shot-launched vertically up
    - Problem on a projectile moving vertically up and down
    - Problem on an arrow shot vertically upward
    - Problem on a ball thrown vertically up from the top of a tower
    - Problem on a toy rocket launched vertically up from a tall platform
    - A soccer ball - write the height equation in vertex form
    - A flare is launched from a life raft vertically up
    - A tangled problem on a ball thrown upward
    - OVERVIEW of lessons on a projectile thrown/shot/launched vertically up

My other lessons on a projectile thrown/shot/launched horizontally or at an angle to horizon are
    - Problems on a projectile thrown horizontally
    - Miscellaneous problems on projectiles

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

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