Question

A cart of mass m1 = 11 kg slides down a frictionless ramp and is made to collide with a second cart of mass m2 = 24 kg which then heads into a vertical loop of radius 0.25 m (a) Determine the height h at which cart #1 would need to start from to make sure that cart #2 completes the loop without leaving the track. Assume an elastic collision. (b) Find the height needed if instead the more massive cart is allowed to slide down the ramp into the smaller cart.

Answer 1

m1 = 11 kg   m2 = 24 kg

speeds before collision

u1 = sqrt(2*g*h) m/s     u2 = 0 m/s

speeds after collision

v1 = ?        v2 = sqrt(5gr) =sqrt(5*9.8*0.25) = 12.25 m/s

initial momentum before collision

Pi = m1*u1 + m2*u2

after collision final momentum

Pf = m1*v1 + m2*v2

from moentum conservation

total momentum is conserved

Pf = Pi

m1*u1 + m2*u2 = m1*v1 + m2*v2 .....(1)

from energy conservation

total kinetic energy before collision = total kinetic energy after collision

KEi = 0.5*m1*u1^2 + 0.5*m2*u2^2

KEf =   0.5*m1*v1^2 + 0.5*m2*v2^2

KEi = KEf

0.5*m1*u1^2 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2

solving 1 & 2

v1 = [(m1-m2)*u1 + (2*m2*u2)]/ (m1+m2)

v2 = [(m2-m1)*u2 + (2*m1*u1)] / (m1+m2)

12.25 = (2*11*sqrt(2*9.8*h))/(11+24)

h = 19.4 m    <<<<<..........answer

m1 = 24 kg   m2 = 11 kg

speeds before collision

u1 = sqrt(2*g*h) m/s     u2 = 0 m/s

speeds after collision

v1 = ?        v2 = sqrt(5gr) =sqrt(5*9.8*0.25) = 12.25 m/s

initial momentum before collision

Pi = m1*u1 + m2*u2

after collision final momentum

Pf = m1*v1 + m2*v2

from moentum conservation

total momentum is conserved

Pf = Pi

m1*u1 + m2*u2 = m1*v1 + m2*v2 .....(1)

from energy conservation

total kinetic energy before collision = total kinetic energy after collision

KEi = 0.5*m1*u1^2 + 0.5*m2*u2^2

KEf =   0.5*m1*v1^2 + 0.5*m2*v2^2

KEi = KEf

0.5*m1*u1^2 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2

solving 1 & 2

v1 = [(m1-m2)*u1 + (2*m2*u2)]/ (m1+m2)

v2 = [(m2-m1)*u2 + (2*m1*u1)] / (m1+m2)

12.25 = (2*24*sqrt(2*9.8*h))/(11+24)

h = 4.07 m    <<<<<..........answer