Question

A 45.0-kg girl is standing on a 152-kg plank. The plank, originally at rest, is free to slide on a frozen lake, which is a flat, frictionless surface. The girl begins to walk along the plank at a constant velocity of 1.55 m/s to the right relative to the plank. (Let the direction the girl is moving in be positive. Indicate the direction with the sign of your answer.) (a) What is her velocity relative to the surface of ice? m/s (b) What is the velocity of the plank relative to the surface of ice? m/s

Answer 1

Given: Mass of girl, M_g = 45Kg ; Mass of planck, M_p = 152Kg ; Initial velocity of Plank, u_p= 0 (at rest) ; Initial velocity of girl, u_g= 0 (at rest) ; Velocity of girl relative to plank, V_gp= 1.55m/s ; the right side direction in which the girl is walking is chosen to be positive.

[All units and calculations in S.I units wherever not shown.]

Required to find: (i) velocity of girl relative to ice, v_gi, (ii) velocity of plank relative to ice, v_pi

Diagram:

Fig. 1

Working formula: We use the concept of conservation of momentum to solve the problem. According to conservation of momentum, the summation of initial momentums of the system equals summation of the final momentums.

Therefore, , M₁, M₂ are two masses, u₁, u₂ are their initial velocities and v₁, v₂ are their final velocities respectively.

Solution:

(a)

The velocity of the girl relative to the surface of ice can be written as the velocity of girl with respect to plank plus the velcoity of plank with respect to ice. [From the relative velocity formula: V_AC=V_AB + V_BC]

Therefore, . The value of v_gp is given to be 1.55m/s.

-----(1)

From the conservation of linear momentum, we have,

-----(2)

[Since the girl and the plank together constitute a system, we take their velocity relative to their surrounding, that is, the ice.] Since initially the system is at rest, the LHS of the equation becomes zero [u=0].

Substituting the values,

-----(3)

Substituting the value of vp_i from equation (1) to equation (3),

Answer: The value of v_gi, velocity of the girl relative to the ice is 1.195 m/s.

(b)

Putting the value of v_gi in equation (1),

Answer: The value of v_pi, the velocity of plank relative to ice is -0.355m/s. The negative sign actually indicates the direction opposite to the chosen positive direction, i.e, the plank actually moves left as the girl walks towards right.

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