Question

The track shown in the figure is frictionless. The block B of mass 4 kg is lying at rest and block A of mass 2 kg is pushed along the track with some speed. The collision between A and B is perfectly elastic.With what velocity should the block A start such that block B just reaches at point P?

• A. (A) 10 m/s
• B. (B) 5 m/s
• C. (C) 20 m/s
• D. (D) 15 m/s

Answer 1

Answer: (A)

(A) 10 m/s

Answer 2

Explanation:
Given:Mass of the block A, mA = 2 kgMass of the block B, mB = 4 kgHeight of block A from the ground, = 4 mThe collision between block A and block B is perfectly elastic.We have to find the velocity with which the block A should start such that block B just reaches the point P.Let the block A starts with velocity uA and it reaches to block B with the velocity vA.Applying the law of conservation of energy for block A, we have12mAuA2+mAgh = 12mA vA2⇒μ2A2+10×4=v2A2⇒μA2+802=vA22⇒vA2 = μA3 +80....(i)This is the velocity with which block A collides with the block B.Applying the law of conservation of linear momentum just before and after the collision, we havemAvA+0=mAvA+mBuB,[Initial momentum of the block B is zero as it is at rest.]where,VA : final velocity of the block AUB : velocity of the block B after the collision⇒2vA = 2VA = 4uB⇒vA = 2uB+VA .......(ii)Now, as the collision between the blocks A and B is elastic, applying the law of conservation of kinectic energy just before and after the collision. We have,12mAvA2=12mAvA2+12mBuB2⇒vA2=2uB2+VA2.....(iii)Substituting vA from eq. (iii) in eq. (ii), we getvA=2uB+vA2−2uB2⇒vA−2uB=vA2−2uB2⇒vA2+4uB2−4vAuB=vA2−2uB2 [squaring both sides]⇒4uB2+2uB2=4vAuB⇒6uB2=4vAuB⇒6uB=4vA⇒uB=23vA.....(iv)Substituting vA from eq. (i) in eq. (iv), we getuB=23uA2+80......(v)This is the velocity of the block B with which it moves after the collision.To reach just at the point P, all of the kinetic energy of the block B gets converted into its poteintial energy .⇒12mBuB2=mBgh⇒uB2=2gh⇒49(uA2+80)=2×10×4 .......[using eq .(v)]⇒uA2+80=2×10×4×94⇒uA2=100m2/s2⇒uA=10m/sThis is the velocity with which the block A should start.Hence, the correct option is (A).