A particle A is projected from the ground with an initial velocity of 10 m/s at an angle of 60 degree with horizontal. From what height h should an another particle B be projected horizontally with velocity 5 m/s so that both the particles collide in ground at point C if both are projected simultaneously (g=10 m/s²)

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Solution

$W h e n a p a r t i c l e i s p r o j e c t e d w i t h s p e e d u = 10 m s^{- 1} m a k i n g a n a n g l e o f 60 ° w i t h t h e g r o u n d, t h e h o r o z o n t a l v e l o c i t y = u \cos 60 ° = 10 \times \frac{1}{2} = 5 m s^{- 1} a n d v e r t i c a l v e l o c i t y u p = u \sin 60 ° = 10 \times \frac{\sqrt{3}}{2} m s^{- 1} . T h e v e l o c i t y a t h i g h e s t p o i n t = 0 a n d c o r r e s p o n d i n g h i g h t a t t a i n e d = h . T h e r e f o r e, f r o m t h i r d e q u a t i o n o f m o t i o n f o r v e r t i c a l m o t i o n i s g i v e n b y, v^{2} = u^{2} - 2 g h . A t t h e h e i g h e s t p o i n t v = 0, i n i t i a l v e l o c i t y u = u \sin θ, t h e r e f o r e, 0 = {(u \sin θ)}^{2} - 2 g h . i . e . 0 = (10 \times \frac{\sqrt{3}}{2})^{2} - 2 \times 10 \times h, w h e r e g = 10 m s^{- 2} . o r h = 100 \times \frac{3}{4 \times 2 \times 10} o r h = \frac{15}{4} o r h = 3.75 m . T h e h o r i z o n t a l v e l o c i t y t h r o u g h o u t t h e p r o j e c t i l e m o t i o n r e m a i n s t h e s a m e, t h e r e f o r e w h e n t h e p a r t i c l e i s a t t h e h e i g h t o f 3.75 m f r o m g r o u n d, t h e t i m e t a k e n t o r e a c h t h e r e i s g i v e n b y, 0 = u \sin θ - g t o r 10 \times \frac{\sqrt{3}}{2} - 10 t = 0 o r t = 1.732 s . W h e n s e c o n d b o d y t h r o n e h o r i z o n t l y, i t s f a l l i s f r e e, t h e r e f o r e i n 1.732 s i t w i l l f a l l b y d i s \tan c e' h_{0} = \frac{1}{2} \times 10 \times {(\sqrt{3})}^{2} = 15 m . T h e r f o r e s e c o n d p a r t i c l e m u s t b e t h r o w n s i m u l \tan e o u s l y f r o m a h e i g h t o f (3.75 + 15) m = 18.75 m . N o t e t h a t h o r i z o n t a l v e l o c i t y i s i n d e p e n d e n t o f v e r t i c a l v e l o c i t y .$

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