A particle when projected vertically upwards from the ground takes time T to reach maximum height H .find out
a) if particle crosses a point at height 7H/ 16 at time t1 and t2 then find t1/t2?
Answers
A particle when projected vertically upwards the grounds takes time T to reach maximum height H.
Let initial velocity of particle is u
final velocity of particle (velocity at maximum height) , v = 0
so, using formula, v² = u² + 2aS
here , v = 0, a = -g and S = H
so, 0 = u² - 2gH
or, H = u²/2g ......(1)
and time taken to reach maximum height is T
so, v = u + at
or, 0 = u - gT => T = u/g.....(2)
now, using formula S = ut + 1/2 at² for other two cases.
7H/16 = ut1 + 1/2(-g)t1²
or, 7H/16 = ut1² - 1/2gt1² ....(3)
similarly, 7H/16 = ut2 - 1/2gt2² .....(4)
from equations (3) and (4) it seems that t1 and t2 are roots of quadratic equations 7H/16 = ut - 1/2gt² or 8gt² - 16ut + 7H = 0
so, sum of roots = t1 + t2 = 16u/8g = 2u/g
from equation (2),
t1 + t2 = 2T..........(5)
now product of roots = t1.t2 = 7H/8g
from equations (1),
t1.t2 = 7u²/16g²
= (7/16)(u/g)²
[ from equation (2), T = u/g]
t1.t2= (7/16)T² ......(6)
now, can we find (t2 - t1) from equations (5) and (6),
t2 - t1 = √{(t1 + t2)² - 4t1.t2}
= √{4T² - 4 × 7/16T²}
= √{16T² - 7T²}/2
= 3T/2 = .....(7)
now from equations (5) and (7),
t2 = (7/4)T
t1 = (1/4)T
so, ratio of t1 and t2 = (1/4) : (7/4) = 1 : 7
Answer:
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