NUMERICAL.
A vehicle of mass 30 quintals moving with a speed of 18km/hr collides with another vehicle of mass 90 quintals moving with a speed of 14.4 km/hr. What will be the velocity of each after the collision if they are in opposite directions?
let the initial and final velocities of 30 quintal body be u and v while that of 90 quintal body be u' and v' respectively.
here we shall use the laws of conservation of energy and momentum
so,
mu + m'u' = mv + m'v'..........................................(1)
and
law of conservation of (kinetic) energy
(1/2)mu2 + (1/2)m'u'2 = (1/2)mv2 + (1/2)m'v'2
or
mu2 + m'u'2 = mv2 + m'v'2....................................(2)
now, here
m = 30 quintals = 3000 kg
m' = 90 quintals = 9000 kg
u = 18 km/hr = 5 m/s
u' = 14.4 km/hr = 4 m/s
thus, equation 1 becomes
3000x5 + 9000x4 = 3000v + 9000v'
or dividing by 3000 on both sides
5 + 12 = v + 3v'
or
v + 3v' = 17...........................................................(3)
or v = 17 - 3v'........................................................(4)
now, equation (2) becomes
3000x52 + 9000x42 = 3000v2 + 9000v'2
or
25 + 48 = v2 + 3v'2
or
v2 + 3v'2 = 73..........................................................(5)
now, by substituting the value of v from (4) in (5), we get
(17 - 3v')2 + 3v'2 = 73
or
9v'2 + 289 - 102v' + 3v'2 = 73
or
12v'2 - 102v' + 216 = 0
dividing both sides by 6, we have
2v'2 - 17v' + 36 = 0
now, by solving the above quadratic equation given us two values of v'
that is
v' = 4 m/s or v' = 4.5 m/s
we can substitute both value of v; one by one in (3) to get two corresponding value of v
so,
(a)
if v' = 4m/s ; v = 5 m/s
and
(b) if v' = 4.5 m/s = 3.5 m/s
we shall select option (b) to be the correct values as option (a) represents the initial velocities.
So, the final velocities will be
v = 3.5 m/ and v' = 4.5 m/s