Welcome to NRICH.

 
Particle projection
By Philip Ellison on Thursday, April 04, 2002 - 11:05 am:

Could someone please help me with the following two questions:

1. A particle is projected under the influence of gravity from a point O on a level plane in such a way that, when its horizontal distance from O is c, its height is h. It then lands on the plane at a distance c+d from O. Show that the angle of projection satisfies
tan a=h(c+d)/cd
The second part of this question involves deriving the conditions that the speed of projection must satisfy, but if someone could help me with the first part then I should be able to sort this out.

2. A particle is projected from a point O with speed root(2gh), where g is the accn. due to gravity. Show that it is impossible, whatever the angle of projection for the particle to reach a point above the parabola
x2=4h(h-y)
where x is horizontal distance from O and y is vertical.
I have gotten to a point where I have to prove that
xtana - (x2)/4hcos a is less than h-(x2)/4h,
however, I can't prove this and may be doing the completely wrong thing!
Any help would be much appreciated
Thanks

By Kerwin Hui on Thursday, April 04, 2002 - 12:25 pm:

Philip,

Q1: Under the usual notation:
(1) c=vt cos a
(2) h=vt sin a-gt2/2
(3) c+d=v2sin 2a/g

Eliminate t (from (1) and (2)) gives:
(4) h=c tan a - (gc2/2v2) sec2a

Eliminate v (from (3) and (4)) gives:
(5) h=c tan a - [c2/2(c+d)] sin 2a sec2 a= {c - [c2/(c+d)]} tan a

which gives the required answer.

For Q2, suppose the point (x,y) is on the trajectory with projection angle a. Then we obtain

y = x tan a - (x2/4h) sec2 a = -x2/4h + x tan a - (x2/4h) tan2 a

So

x2 tan2 a - 4hx tan a + (4hy+x2) = 0

Condition for no real roots gives (2hx)2<x2(4hy+x2). Rearranging gives the answer stated.

Kerwin

By Philip Ellison on Thursday, April 04, 2002 - 03:33 pm:

Thanks Kerwin, I didn't think about deriving (3) in the first question and forgot that the second one could be rearranged into a quadratic in tan alpha. Hopefully I'll have improved by the time I come to take my STEP exams (if I take them at all)!