Activity 1.4 - Calculations on the modeling of the ballpoint case

Learning Aims:
  • Students use calculations to be able to decide better which of the scenarios are possible.
Materials:
  • Pen, paper, calculator
  • Program Coach (optional)
Suggestions for use:
  • Introduce the physics concepts necessary for this calculation in a classroom discussion.
  • In order to prepare the students for the next activity, focus on inertia.
  • The first part of this activity is a rather ‘closed’ activity to make sure that all the students are at the same level.
  • In the second part the students have to discuss the probability of both the scenarios. They have to compare the results from different experiments with each other.
  • Let the students discuss how valuable, objective and discerning their results are.

Examples of calculations:

Pen dropping on a mandarin

  • The height from which the pen starts is 100 cm, the mass of the pen is 11 g, the pen penetrates the mandarin for 0,5 cm. Calculate the force between pen and mandarin during the collision.

    • First put all the data in SI-units: h = 1.00 m ; m = 0.011 kg ; d = 0.005 m
    • The speed that the pen has when making first contact with the mandarin ( we neglect friction) can be derived from: E = mgh = ½ m v2
    • v2 = 2 * g * h = 2 * 9.8 * 1 = 19.6 (m/s)2
    • v = 4.4 m / s
  • The starting velocity is v = 4.4 m / s, the stopping distance is 0.005 m. We use W = F * s for the work done, this work is equal to the kinetic energy of the ballpoint at the beginning of the collision.
  • Ek = ½ m * v2 = 0.5 * 0.011 * 19.6 = 0.108 J
  • F = W / s = 0.108 / 0.005 = 21.6 N

Mandarin dropping on a pen

  • We now take a mandarin (100 g) and assume that the force between mandarin and pen is a constant 21.6 N. We then calculate how far the pen would penetrate the mandarin when the mandarin would drop from a height of 1.00 m.

    • The starting velocity is v = 4.4 m / s (as follows from the previous calculation), the force is 21.6 N. We use W = F * s to calculate the stopping distance.
    • The kinetic energy of the mandarin is Ek = ½ m * v2 = 0.5 * 0.100 * 19.6 = 0.98 J

The stopping distance s is s = W / F = 0.98 / 21.6 = 0.045 m

Possible questions:
  • Calculations with other data (height, weight).
  • Derive a general formula for two objects (with masses m1 and m2) falling from the same height, to calculate the penetration distance s.
    • Using the same steps as in the former calculation, we find

m1 * g * h1 = F * s1 and for the other case m2 * g * h2 = F * s2

We can write F = ( m1 * g * h1 ) / s1

and thus find: m2 * g * h2 = {( m1 * g * h1 ) / s1 }* s2

In this specific question h1 = h2 and g is also the same in both cases. This means:

m2 = { m1 / s1 }* s2 and this can be written as: m2 / s2 = m1 / s1 or m1* s2 = m2 * s1

  • Derive a general formula for two objects (with equal masses) colliding with different velocities, to calculate the penetration distance s.

    • o This time we calculate the kinetic energy, this is equal to the work done while stopping the object. ½ m v2 = F * s
    • o ½ m1 v12 = F * s1 and ½ m2 v22 = F * s2
    • Since F is equal for both cases, we find: ½ m1 v12 / s1 = ½ m2 v22 / s2
    • For equal objects (m1 = m2) we then find: v12 / s1 = v22 / s2