A projectile is shot from the ground at an angle of #pi/4 # and a speed of #8 m/s#. Factoring in both horizontal and vertical movement, what will the projectile's distance from the starting point be when it reaches its maximum height?

2 Answers
May 29, 2017

The distance is #=3.65m#

Explanation:

Resolving in the vertical direction #uarr^+#

initial velocity is #u_y=vsintheta=8*sin(1/4pi)#

Acceleration is #a=-g#

At the maximum height, #v=0#

We apply the equation of motion

#v=u+at#

to calculate the time to reach the greatest height

#0=8sin(1/4pi)-g*t#

#t=8/g*sin(1/4pi)#

#=0.577s#

The greatest height is

#h=(8sin(1/4pi))^2/(2g)=1.63m#

Resolving in the horizontal direction #rarr^+#

To find the horizontal distance, we apply the equation of motion

#s=u_x*t#

#=8cos(1/4pi)*0.577#

#=3.27m#

The distance from the starting point is

#d=sqrt(h^2+s^2)#

#=sqrt(1.63^2+3.27^2)#

#=3.65m#

May 29, 2017

Please check the animations carefully. The answer is given at the end of the explanation.

Explanation:

#"Projectile motion is a special case of two-dimensional motion. "#
#"An object found at the moment of shooting is seen."#

enter image source here

#"if we ignore the air resistance and gravity,the object move to infinity."#
#"But that's not really the case. The object is attracted by the ground"##" and placed on a trajectory."#
#"The object drops freely from point J to point H."#

enter image source here

#"We must divide the motion horizontally and vertically into two parts."#

  • The blue vector shows the horizontal component of the initial velocity.

enter image source here

#"The " V_x " vector can be calculated using " v_x=vi*cos theta#

  • The magnitude and direction of blue vector does not change.
  • *The blue vector allows the object to move horizontally. *

enter image source here

  • The green vector shows the vertical component of the initial velocity.

enter image source here

#"The " V_y " vector can be calculated using " v_y=vi*sin theta#

#V_y " at any time is "v_y=v_i*sin theta -g*t #

enter image source here

  • The magnitude and direction of green vector change.
  • *The blue vector allows the object to move vertically. *
  • the magnitude of green vector at maximum height is zero.

#"How can I calculate the elapsed time from initial point to peak point ?"#

#v_y=0("at maximum height)"#
#0=v_i*sin theta-g*t#
#v_i*sin theta=g*t#
#t=(v_i*sin theta)/g#

#"How can I calculate the maximum height?"#

enter image source here

#h_m=(v_i^2*sin^2 theta)/(2*g)#

#"How can I calculate the elapsed time for traveled from initial point to end point ?"#

#t=2*(v_i*sin theta)/2#

#"How do we find the velocity of a projectile at any time over its trajectory?"#

enter image source here

#"Calculate "v_x=v_i*cos theta#
#"Calculate "v_y=v_i*sin theta-g*t#
#"Calculate "v=sqrt((v_x)^2+(v_y)^2)#

#"How can I calculate the location of object on trajectory?"#

enter image source here

#x=v_i*t*cos theta#
#y=v_i*t*sin theta-1/2 g t^2#

#"How can I calculate the maximum x-range?"#

#x_m=(v_i^2*sin(2 theta))/g#

#"answer to your question."#
#theta=pi/4#
#v_i=8 m*s^(-1)#

#t=(v_i*sin theta)/g=(8*0.707)/(9.81)=0.58 s#

#x=v_i*t*cos theta=8*0.58*0.707=3.28m#