Question

Twins are sliding a rock back and forth on a frozen lake with negligible friction. The mass of each child is 21.0 kg and the mass of the rock is 2.6 kg. Each time the rock is given a push by the child who just caught it, it leaves the hand with a speed of 1.8 m/s relative to the child's hand.

A. The twins are intitially at rest and the sister first slides the rock toward her brother in the positive direction. What is the speed of the sister after she releases the rock?
m/s
B. The brother receives the rock. What is his velocity after he has received it?
m/s
C. The brother then slides the rock to his sister. What is his velocity after he releases the rock?
m/s
D. The sister catches the rock. What is her velocity now?
m/s
E. Can the game go on forever?

a.) no

b.) Cannot be determined    

c.) yes

Answer 1

A)

For the sister :

m = mass of the rock = 2.6 kg

v_rs = velocity of rock relative to sister = 1.8 m/s

v_rg = velocity of rock relative to ground

v_sg = velocity of sister relative to ground

M = mass of sister = 21 kg

we know that :

v_rs = v_rg - v_sg

1.8 = v_rg - v_sg

v_rg = 1.8 + v_sg   

Using the conservation of energy

M v_sg = - m v_rg

(21) v_sg = - (2.6) (1.8 + v_sg)

v_sg = - 0.1983 m/s

Speed of sister = 0.1983 m/s

b)

v_rg = 1.8 + v_sg

v_rg = 1.8 + (- 0.1983)

v_rg = 1.602 m/s

v_bg = velocity of brother after receiving the rock

Using conservation of momentum

m  v_rg = (M + m) v_bg

(2.6) (1.602) = (21 + 2.6) v_bg

v_bg = 0.1765 m/s

c)

m = mass of the rock = 2.6 kg

v_rgi = velocity of rock relative to ground initially = 0.1765 m/s

v_bgi = velocity of brother relative to ground initially = 0.1765 m/s

v_rbf = velocity of rock relative to brother finally = - 1.8 m/s

v_rgf = velocity of rock relative to ground finally

v_bgf = velocity of brother relative to ground finally

M = mass of brother = 21 kg

we know that :

v_rbf = v_rgf - v_bgf

- 1.8 = v_rg - v_bgf

v_rgf = - 1.8 + v_bgf     

Using the conservation of energy

M v_bgi + m v_rgi = M v_bgf + m v_rgf

(21) (0.1765) + (2.6) (0.1765) = (21) v_bgf + (2.6) (- 1.8 + v_bgf )

v_bgf = 0.375 m/s

d)

v_rgf = - 1.8 + v_bgf = - 1.8 + 0.375 = - 1.425 m/s

Consider the collision between rock and sister

m = mass of the rock = 2.6 kg

v_rgi = velocity of rock relative to ground initially = - 1.425 m/s

v_sgi = velocity of sister relative to ground initially = - 0.1983 m/s

v = velocity of rock and sister together

M = mass of sister = 21 kg

Using the conservation of energy

M v_sgi + m v_rgi = (M + m) v

(21) (- 0.1983) + (2.6) (- 1.425) = (21 + 2.6) v

v = - 0.333 m/s