Worksheet: Newton's Second Law: Variable Mass

In this worksheet, we will practice using differentiation for Newton's second law of motion of a particle with variable mass.

Q1:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠=ο€Ή6𝑑+9π‘‘ο…οŠ¨m. Its mass varies with time such that π‘š=(8𝑑+9)kg. Write an expression for the force acting on the body at time 𝑑.

  • A ( 3 8 4 𝑑 + 1 8 0 ) N
  • B ( 9 6 𝑑 + 1 0 8 ) N
  • C ( 1 9 2 𝑑 + 1 8 0 ) N
  • D ( 9 6 𝑑 + 1 8 0 ) N

Q2:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠=ο€Ή2𝑑+5𝑑+4ο…οŠ¨m. Its mass varies with time such that π‘š=(6𝑑+5)kg. Determine the force acting upon the body when 𝑑=3s.

Q3:

A ball of mass 5 g was moving in a straight line through a medium loaded with dust. The dust was accumulating on its surface at a rate of 1 g/s. Find the magnitude of the force acting on the ball at time 𝑑=5seconds, given that the displacement of the ball is expressed by the relation sc(𝑑)=ο€Ό23𝑑+𝑑+7𝑑+1, where c is a unit vector in the direction of the motion and the displacement is measured in centimeters.

Q4:

The mass of a body at time 𝑑 is given by π‘š(𝑑)=(2𝑑+12)kg, whereas its position vector is rc(𝑑)=ο€Ή2𝑑+3𝑑+15ο…οŠ¨, where c is a constant unit vector, r is measured in meters, and 𝑑 is measured in seconds. Find the magnitude of the force acting on the body at 𝑑=2seconds.

Q5:

A metal ball of mass 220 g was moving in a straight line at 10 m/s through a dusty medium. If it was free from dust at the start of its motion, and the dust adhered to its surface at rate of 0.06 g/s, find the mass of the ball π‘š and the force 𝐹 acting on it at any given time 𝑑.

  • A π‘š = 2 2 0 g , 𝐹 = 6 0 d y n e s
  • B π‘š = 2 2 0 + 0 . 0 6 𝑑 g , 𝐹 = 6 0 d y n e s
  • C π‘š = 2 2 0 + 0 . 0 6 𝑑 g , 𝐹 = 0 . 6 d y n e s
  • D π‘š = 2 2 0 𝑑 + 0 . 0 6 g , 𝐹 = 0 . 6 d y n e s

Q6:

A ball of mass 9 g was moving in a straight line on a dusty plane. The dust was accumulating on the ball’s surface at 2 g/s. The ball’s displacement is expressed by the relation sc(𝑑)=𝑑+3𝑑+7𝑑+2ο…οŠ©οŠ¨, where c is the unit vector in its direction of motion, 𝑑 is the time in seconds, and the magnitude of the displacement is measured in centimeters. Determine the force vector at time 𝑑, given that its magnitude is measured in dynes.

  • A ( 1 2 𝑑 + 1 2 ) c
  • B ο€Ή 1 8 𝑑 + 7 8 𝑑 + 7 2   c
  • C ο€Ή 1 8 𝑑 + 7 8 𝑑 + 6 8   c
  • D ( 5 4 𝑑 + 5 4 ) c
  • E ο€Ή 6 𝑑 + 3 9 𝑑 + 6 8 𝑑 + 6 3    c

Q7:

A ball of mass 22 g was moving in a straight line through a medium loaded with dust. The dust was accumulating on its surface at a rate of 2 g/s. Find the magnitude of the force acting on the ball at time 𝑑=2seconds, given that the displacement of the ball is expressed by the relation sc(𝑑)=ο€Ό13𝑑+2𝑑+𝑑+8, where c is a unit vector in the direction of the motion and the displacement is measured in centimeters.

Q8:

The mass of a body at time 𝑑 is given by π‘š(𝑑)=(3𝑑+10)kg, whereas its position vector is rc(𝑑)=ο€Ή2𝑑+2𝑑+13ο…οŠ¨, where c is a constant unit vector, r is measured in meters, and 𝑑 is measured in seconds. Find the magnitude of the force acting on the body at 𝑑=2seconds.

Q9:

A metal ball of mass 104 g was moving in a straight line at 6 m/s through a dusty medium. If it was free from dust at the start of its motion, and the dust adhered to its surface at rate of 0.08 g/s, find the mass of the ball π‘š and the force 𝐹 acting on it at any given time 𝑑.

  • A π‘š = 1 0 4 g , 𝐹 = 4 8 d y n e s
  • B π‘š = 1 0 4 + 0 . 0 8 𝑑 g , 𝐹 = 4 8 d y n e s
  • C π‘š = 1 0 4 + 0 . 0 8 𝑑 g , 𝐹 = 0 . 4 8 d y n e s
  • D π‘š = 1 0 4 𝑑 + 0 . 0 8 g , 𝐹 = 0 . 4 8 d y n e s

Q10:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠=ο€Ή3𝑑+8𝑑+7ο…οŠ¨m. Its mass varies with time such that π‘š=(5𝑑+4)kg. Determine the force acting upon the body when 𝑑=4s.

Q11:

A body moves in a straight line. At time 𝑑 seconds, its displacement from a fixed point is given by 𝑠=ο€Ή9𝑑+7π‘‘ο…οŠ¨m. Its mass varies with time such that π‘š=(5𝑑+8)kg. Write an expression for the force acting on the body at time 𝑑.

  • A ( 3 6 0 𝑑 + 1 7 9 ) N
  • B ( 9 0 𝑑 + 1 4 4 ) N
  • C ( 1 8 0 𝑑 + 1 7 9 ) N
  • D ( 9 0 𝑑 + 1 7 9 ) N

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