9  Problems 1

Week 2

These problems sheets are designed to help you learn mechanics. You get better at physics by solving problems. You will attempt these questions at the problems class. This means you do not need to complete the problems beforehand, but it might be helpful to have looked over them. Your solutions will not be marked. Solutions will be provided on Blackboard the week following your problems class. If you are using these for revision try to do the questions first without looking at the answers.

9.1 Projectile motion

A particle is launched with speed \(u\) at an angle \(\theta\) to the horizontal. The particle experiences a constant acceleration in the vertical direction due to gravity, \(g\).

  1. What are the \(x\) and \(y\) components of the particle’s initial velocity?
  2. What is the maximum height of the projectile?
  3. How long does it take the particle to return to the ground?
  4. What is the range of the particle?
  5. Differentiate your answer to part (d) to show that the maximum range of the particle is achieved when \(\theta = 45^\circ\).

9.2 Vector addition

Vectors whose moduli are 3, 4, and 6 act in directions making angles \(30^\circ\), \(90^\circ\), and \(135^\circ\) respectively with the positive \(x\)-axis, and all lie in the \(x - y\) plane. Find their sum. Give the modulus and direction of the resultant vector.

  • One way to do this is to work out the \(\vec{\imath}\) and \(\vec{\jmath}\) components of each vector, then add them together.
  • “Modulus” is another word for “magnitude”.

9.3 Angles between vectors

The resultant, \(\vec{R}\), of two intersecting forces \(\vec{F}_a\) and \(\vec{F}_b\) has magnitude \(\sqrt{3} \text{ N}\). If \(|\vec{F}_a| = 1 \text{ N}\) and \(|\vec{F}_b| = 2 \text{ N}\), find the angle between \(\vec{F}_a\) and \(\vec{F}_b\), and the angle between \(\vec{F}_a\) and \(\vec{R}\).

  • Draw a diagram.
  • Use the cosine rule, and the sine rule.

9.4 Non-uniform acceleration with vectors & derivatives

A particle of mass \(m = 2 \text{ kg}\) is acted on by a force \(\vec{F}\) in newtons. The position \(\vec{r}\) of the particle is found to follow a path due to this force given by \[ \vec{r} = \left( \frac{5t^2}{3b} - \frac{t^3}{4c} \right) \hat{\imath} + \left( \frac{3t^2}{b} - \frac{7t}{a} \right) \hat{\jmath} \text{ m} \] where \(a\), \(b\) and \(c\) are constants with the value 1 in appropriate units, when distance is measured in metres and time in seconds. Find the following.

  1. The units of the constants \(a\), \(b\) and \(c\).
  2. The value of \(t\) when the particle is moving parallel to the vector \(\hat{\imath}\).
  3. The force \(\vec{F} = m \vec{a}\), where \(\vec{a}\) is the acceleration, after 5 seconds.
  4. The magnitude of the force determined in part (c).

9.5 Centre of mass of system of particles in 3D

Particles 1,2,3 with mass \(m_1 = 3 \text{ kg}\), \(m_2 = 6 \text{ kg}\) and \(m_3 = 7 \text{ kg}\) are located at positions defined by the vectors \(\vec{r}_1 = 5\hat{\imath} - 7\hat{\jmath} \text{ m}\), \(\vec{r}_2 = 3\hat{\imath} + 6\hat{\jmath} \text{ m}\) and \(\vec{r}_3 = 7\hat{\imath} + 3\hat{\jmath} \text{ m}\).

  1. Determine the position of the centre of mass of the three-particle system.
  2. The particles are subsequently displaced in the direction of \(\hat{k}\) so that their new positions are given by \(\vec{r}^\prime_1 = 5\hat{\imath} - 7\hat{\jmath} + 3\hat{k} \text{ m}\), \(\vec{r}^\prime_2 = 3\hat{\imath} + 6\hat{\jmath} + 6\hat{k} \text{ m}\) and \(\vec{r}^\prime_3 = 7\hat{\imath} + 3\hat{\jmath} - 3\hat{k} \text{ m}\). Determine the new position of the centre of mass of the particle system.
  3. If this translation occurs uniformly over a time \(t\), what is the velocity of the centre of mass frame?

9.6 Bonus question: Galilean transformation using vectors

Galileo reputedly tested his ideas about Galilean relativity and gravity by dropping objects from the leaning tower of Pisa, which has height \(56 \text{ m}\).

  1. Assume Galileo dropped a book of mass \(2.4 \text{ kg}\) from the tower. Write down a vector equation for the book’s position as a function of time in Galileo’s frame, i.e. take Galileo to be the origin, and a gravitational force proportional to \(g\) in the \(-\hat{k}\) direction, and ignore the effects of wind resistance.
  2. A bird was flying near Galileo as he dropped the book, with instantaneous position \(3.1 \hat{\imath} + 2.0\hat{k} \text{ m}\) and velocity \(1.2 \hat{\imath} + 0.6 \hat{\jmath} \text{ m s}^{-1}\) in Galileo’s frame. Write down the time-dependent position of the book in the bird’s frame. What is the position vector, in the bird’s frame, that the book hits the ground?

You might find this much easier to solve if you draw a diagram with vectors showing the positions of the book and the bird relative to Galileo at time \(t\).