Projectile Question Help (1 Viewer)

[iheartOJ]

Hi!

I'm having a little trouble with this question from the Cambridge 3U textbook (Q10 ex. 3H). Can anyone help with part c of this question?

A particle is projected from the origin with velocity V m/s at an angle of a to the horizontal.

a) Assuming that the coordinates of the particle at time t are (Vtcosa, Vtsina - 1/2gt², prove that the horizontal range R of the particle is [V²sin2a]/g
b) Hence prove that the path of the particle has equation y = x (1 - x/R) tana
c) Suppose that a=45^o and that the particle passes through two points 6m apart and 4m above the point of projection. Let x₁ and x₂ be the x-coordinates of the two points.

i) show that x₁ and x₂ are the roots of the equation x² - Rx + 4R​

ii) Use the identity (x₁ - x₂)² = (x₁ - x₂)² - 4x₁x₂ to find R​

Thank you so much!

 

[HeroicPandas]

The diagram is the key to this question, i THINK they provide u with one

If not, draw a normal diagram for a particle projected from the origin

After that mark a dotted (horizontal line) of y = 4 (this is the height of points (or walls?)

At the points of intersection of the horizontal line, y = 4, and the parabola, draw vertical lines down perpendicular to the x-axis (this will look like 2 'sticks' which are 6 metres horizontally apart)

Let the closest 'stick' to the origin be x_1 metres away from origin and the other stick be x_2 metres away from the origin (this is just naming the positions of the sticks/walls or whatever)

c) (i)When u solve the horizontal line with the trajectory of the particle (that u've found), u will see 2 points of intersections, u notice there is 2 values for x (that is x_1 and x_2) this means, when u sub y = 4 into that trajectory of the partcle and solve for 'x', u will certainly get x_1 and x_2 as roots (this occurs when a = 45, as the question has stated)



(ii) The aim is to find R

We know that the 'sticks' are 6 metres apart, so using simple subtraction of distances



Sub into that identity and u get ur answer (u must also find x_1 + x_2 and (x_1)(x_2) - this is ur job!)

All i hope is that the diagram is given or u can follow my steps on how to draw it (sorry if unclear)