Mathematical modelling in mechanicsEdexcel International A Level Maths: Subtopic test

1. 1
   A ball of mass 0.4 kg is thrown vertically upwards from ground level at 14 m s⁻¹. A student models the ball as a particle and ignores air resistance.

   (a)

   Which statement describes what is assumed when the ball is modelled as a particle?

   [1 mark]

   • AThe ball has mass but its size can be ignored
   • BThe ball has no mass
   • CThe ball has size but cannot rotate
   • DThe ball does not experience a weight force

   (b)

   Using the model, what is the maximum height reached by the ball above the ground?

   [1 mark]

   • A20 m
   • B10 m
   • C1.4 m
   • D5 m

   (c)

   Explain why the model gives a maximum height that is greater than the true maximum height.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A crate is pulled across a horizontal floor by a rope attached to the front of the crate. The crate is modelled as a particle, the rope as a light inextensible string and the floor as smooth.

   (a)

   Which statement follows from modelling the rope as light?

   [1 mark]

   • AThe tension in the rope is zero
   • BThe tension is greater at the crate end than at the pulling end
   • CThe tension is the same at every point along the rope
   • DThe rope cannot exert a force on the crate

   (b)

   Which statement follows from modelling the rope as inextensible?

   [1 mark]

   • AThe tension is the same throughout the rope
   • BThe rope has no mass
   • CThe rope can only pull and never push
   • DThe rope does not stretch, so its length stays constant

   (c)

   State what is assumed about the forces on the crate by modelling the floor as smooth, and state one way in which the real motion would differ from the model.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform rod PQ has mass 8 kg and length 3 m. It is modelled as a rigid rod and is held horizontal by two vertical strings, one attached at each end. The strings are modelled as light and inextensible. Take g = 9.8 m s⁻².

   (a)

   Explain what is meant by modelling the rod as (i) uniform and (ii) rigid. State the distance from P at which the weight of the rod is assumed to act.

   [3 marks]

   (b)

   A block of mass 2 kg is placed at the midpoint of the rod and is modelled as a particle. Find the tension in each string, and state two modelling assumptions about the strings, with the effect of each.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Particles A and B, of masses 3 kg and 5 kg, are attached to the ends of a light inextensible string. The string passes over a fixed pulley, modelled as smooth and light, with A and B hanging vertically on either side. The system is released from rest. Take g = 9.8 m s⁻².

   (a)

   State the consequences for the tension and the motion of modelling (i) the string as light and inextensible and (ii) the pulley as smooth and light. Hence find the acceleration of the system.

   [6 marks]

   (b)

   In an experiment the measured acceleration of the system is 2.1 m s⁻². Calculate the percentage by which the model's value of the acceleration is greater than the measured value. Suggest two refinements to the model and explain the effect of each on the predicted acceleration.

   [6 marks]

   Total for question 4: 12 marks

End of questions