Question

A car of mass 1000kg moving with 18km/hr on a smooth road collide with a horizontal string of a string constant 6.25×10³ n/m . find comprehension of string . In the above question if coefficient of friction is 0.5 . calculate maximum comprehension? ​

Answer 1

mass of car, m = 1000kg

initial speed of car, u = 18km/h = 18 × 5/18 = 5 m/s

so, kinetic energy of car, K.E = 1/2 mu²

= 1/2 × 1000 × (5)²

= 12500 J

the car collides with a horizontal spring.

if car comes to rest, then all kinetic energy of car converted into spring potential energy.

i.e., kinetic energy of car = spring potential energy

or, 12500 = 1/2 × kx² , where x is compression of spring.

or, 12500 = 1/2 × 6.25 × 10³ × x²

or, 4 = x²

or, x = 2 m

hence, compression of spring is 2m.

now, coefficient of friction is u = 0.5

let maximum compression is y.

so, lost in kinetic energy due to friction = u × normal reaction × y

= 0.5 × mg × y

= 0.5 × 1000 × 10 × y

= 5000y

from conservation of energy,

kinetic energy = 5000y + spring potential energy

or, 12500 = 5000y + 1/2 × 6.25 × 10³ × y²

or, 12500 = 5000y + 3125y²

or, 4 = 1.6y + y²

or, y² + 1.6y - 4 = 0

or, y = {-1.6 ± √(2.56 + 16)}/2 ≈ 1.35 , -2.95

but negative value of y doesn't possible.

so, y = 1.35 m