A projectile is shot from the ground at an angle of π4 and a speed of 8ms. 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 +

initial velocity is uy=vsinθ=8sin(14π)

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(14π)gt

t=8gsin(14π)

=0.577s

The greatest height is

h=(8sin(14π))22g=1.63m

Resolving in the horizontal direction +

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

s=uxt

=8cos(14π)0.577

=3.27m

The distance from the starting point is

d=h2+s2

=1.632+3.272

=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 Vx vector can be calculated using vx=vicosθ

  • 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 Vy vector can be calculated using vy=visinθ

Vy at any time is vy=visinθgt

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 ?

vy=0(at maximum height)
0=visinθgt
visinθ=gt
t=visinθg

How can I calculate the maximum height?

enter image source here

hm=v2isin2θ2g

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

t=2visinθ2

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

enter image source here

Calculate vx=vicosθ
Calculate vy=visinθgt
Calculate v=(vx)2+(vy)2

How can I calculate the location of object on trajectory?

enter image source here

x=vitcosθ
y=vitsinθ12gt2

How can I calculate the maximum x-range?

xm=v2isin(2θ)g

answer to your question.
θ=π4
vi=8ms1

t=visinθg=80.7079.81=0.58s

x=vitcosθ=80.580.707=3.28m