Physics 2: free practice, theory and problems

A projectile moves in two directions at once: constant velocity horizontally and free fall vertically. Split the motion into an -part and a -part and calculate them separately.

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Projectiles and motion in two dimensions
Circular motion and gravitation
Electromagnetism
Quantum physics and relativity

1. Projectiles and motion in two dimensions

What is it about?

A projectile moves in two directions at once: constant velocity horizontally and free fall vertically. Split the motion into an $x$ -part and a $y$ -part and calculate them separately.

Concepts and formulas

Initial speed $v_0$ at angle $\alpha$ : $v_{0x} = v_0\cos\alpha$ , $v_{0y} = v_0\sin\alpha$ .
Horizontal: $x = v_{0x}t$ . Vertical: $y = v_{0y}t - \tfrac12 gt^2$ .
Range on level ground: $R = \dfrac{v_0^2\sin 2\alpha}{g}$ (largest at 45°).
Maximum height: $h = \dfrac{v_{0y}^2}{2g}$ .
Horizontal throw from height $h$ : fall time $t = \sqrt{2h/g}$ , distance $x = v_0 t$ .

Horizontal: constant velocity. Vertical: free fall. Time links them together.

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2. Circular motion and gravitation

What is it about?

A body in circular motion needs a force directed towards the centre. For planets and satellites that force is gravity.

Concepts and formulas

Centripetal acceleration $a = \dfrac{v^2}{r}$ , centripetal force $F = \dfrac{mv^2}{r}$ .
Newton's law of gravitation: $F = G\dfrac{m_1m_2}{r^2}$ , with $G = 6.67 \cdot 10^{-11}\ \text{Nm}^2/\text{kg}^2$ .
Orbital speed of a satellite: $v = \sqrt{\dfrac{GM}{r}}$ , where $r$ is measured from the centre of the planet.
Earth's mass $5.97 \cdot 10^{24}$ kg and radius $6.37 \cdot 10^{6}$ m.

Circle: the force points towards the centre. Gravity decreases with the square of the distance.

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3. Electromagnetism

What is it about?

Electricity and magnetism are connected. Current creates magnetic fields, magnetic fields exert forces on moving charges, and a changing magnetic field induces a voltage.

Concepts and formulas

Force on a moving charge: $F = qvB$ (when the velocity is perpendicular to the field).
Force on a current-carrying conductor: $F = ILB$ .
A charge in a uniform magnetic field moves in a circle with radius $r = \dfrac{mv}{qB}$ .
Induced emf in a moving conductor: $\varepsilon = vBL$ . Faraday's law: $\varepsilon = -\dfrac{d\Phi}{dt}$ .
Lenz's law: the induced current opposes the change that caused it.

A change in magnetic flux induces a voltage. That is how generators work.

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4. Quantum physics and relativity

What is it about?

On very small scales light behaves like particles (photons), and particles like waves. At speeds close to the speed of light, clocks run slower for the one who is moving.

Concepts and formulas

Photoelectric effect: $E_k = hf - W$ , where $W$ is the work function. The energy of a photon in eV: $E \approx \dfrac{1240}{\lambda\,(\text{nm})}$ .
de Broglie wavelength: $\lambda = \dfrac{h}{p}$ .
Time dilation: $\Delta t = \gamma\,\Delta t_0$ , with $\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$ .
Length contraction: $L = L_0/\gamma$ . Energy and mass: $E = mc^2$ .

Higher frequency gives more energy per photon.

\gamma

is always 1 or larger.

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Example problems with solutions

Here are some of the problems in physics 2. In the app, calculation problems get new numbers every time, so you can practise until it sticks – and take a graded practice exam before the real one.

Projectiles and motion in two dimensions: At which angle does a throw on level ground give the longest range (without air resistance)?

Answer: 45°

$\sin 2\alpha$ is largest when $2\alpha = 90^\circ$ .

Circular motion and gravitation: The distance between two masses is doubled. What happens to the gravitational force?

Answer: It becomes a quarter

$F \propto 1/r^2$ , so double the distance gives $1/4$ .

Electromagnetism: What does Lenz's law say?

Answer: The induced current opposes the change that caused it

It follows from conservation of energy.

Quantum physics and relativity: Why does red light not release electrons from a metal, whatever the intensity, while UV light does?

Answer: Each red photon has too little energy

It is the energy per photon ( $hf$ ) that must exceed the work function.

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Physics 2: free practice, theory and problems

Contents

1. Projectiles and motion in two dimensions

What is it about?

Concepts and formulas

2. Circular motion and gravitation

What is it about?

Concepts and formulas

3. Electromagnetism

What is it about?

Concepts and formulas

4. Quantum physics and relativity

What is it about?

Concepts and formulas

Example problems with solutions