Exercises: Rectilinear and circular motion

A commuter train travels in a straight line at a constant velocity of $\vec{v} = 20\,\vec{i}\ \text{m/s}$. When $t = 0$, it passes through position $x_0 = 100\ \text{m}$.

1. a) Write the equation for its position $x(t)$.
2. b) Calculate its position when $t = 8\ \text{s}$.

An aircraft starts from rest on the runway and accelerates uniformly with $\vec{a} = 3\,\vec{i}\ \text{m/s}^2$.

1. a) Write the equation for its velocity $v(t)$.
2. b) Calculate its velocity after $10\ \text{s}$.
3. c) Explain in your own words what an acceleration of $3\ \text{m/s}^2$ means.

A cyclist moves in a straight line with uniform acceleration (MRUA), with $x_0 = 0$, $v_0 = 4\ \text{m/s}$ and $a = 1\ \text{m/s}^2$.

1. a) Write its position equation $x(t)$.
2. b) Calculate its position when $t = 6\ \text{s}$ using $x = x_0 + v_0 \times t + \frac{1}{2} \times a \times t^2$.

A coin is dropped from the top of a $20\ \text{m}$ tower. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the time it takes to reach the ground.
2. b) Calculate the velocity with which it arrives.

The blades of a ceiling fan take $0.5\ \text{s}$ to complete one full revolution.

1. a) What are the period $T$ and the frequency $f$ of the rotation?
2. b) Calculate the angular velocity $\omega = \frac{2\pi}{T}$, in $\text{rad/s}$, and round it to one decimal place.

A Ferris wheel rotates with constant angular velocity $\omega = 0.5\ \text{rad/s}$ and its radius is $r = 10\ \text{m}$. The diagram shows one car on the rim and another closer to the axle.

1. a) Calculate the linear velocity $v = \omega \times r$ of a car located on the rim.
2. b) Explain why a car located closer to the axle would move more slowly, even though the angular velocity is the same.

A skater takes a bend of radius $8\ \text{m}$ at $4\ \text{m/s}$. The diagram shows the bend and the velocity vector.

1. a) Calculate their normal acceleration $a_n = \frac{v^2}{r}$.
2. b) State its direction and sense.

A satellite describes an approximately circular orbit around the Earth at constant speed.

1. a) Does the satellite have acceleration?
2. b) If the magnitude of the velocity does not change, what changes about the velocity vector? Explain why this implies that acceleration exists and what this acceleration is called.

A bus travelling at $72\ \text{km/h}$ brakes uniformly at $a = -2\ \text{m/s}^2$ until it comes to a stop.

1. a) Express its initial velocity in $\text{m/s}$.
2. b) Calculate the time it takes to stop.
3. c) Calculate the braking distance.

A stone is thrown vertically downwards at $v_0 = 10\ \text{m/s}$ from a bridge $15\ \text{m}$ high. Take $g \approx 10\ \text{m/s}^2$. Look at the diagram:

1. a) Set up the position equation taking the water surface as the origin and calculate the time it takes to reach the water.
2. b) Calculate the velocity with which it arrives.

A juggler throws a ball vertically upwards at $v_0 = 30\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$. Look at the launch diagram:

1. a) Calculate the time it takes to reach maximum height.
2. b) Calculate this maximum height.
3. c) How much time passes until the ball returns to his hand and with what velocity does it arrive? Justify your answer using the symmetry of the motion.

A tube train starts from rest and accelerates uniformly at $0.5\ \text{m/s}^2$ over $400\ \text{m}$.

1. a) Calculate the velocity it reaches using $v^2 = v_0^2 + 2 \times a \times \Delta x$.
2. b) Calculate the time it takes to cover those $400\ \text{m}$.

A merry-go-round with a radius of $4\ \text{m}$ makes one revolution every $8\ \text{s}$.

1. a) Calculate its angular velocity $\omega$ in $\text{rad/s}$, in terms of $\pi$.
2. b) Calculate the linear velocity of a child riding on the edge.
3. c) Calculate their normal acceleration. Express the final results to one decimal place as well.

A bicycle moves forward at $7\ \text{m/s}$ and its wheels have a radius of $0.35\ \text{m}$.

1. a) Calculate the angular velocity of the wheels, given that at the rim $v = \omega \times r$.
2. b) Calculate the period of rotation of the wheel.
3. c) Calculate the distance the bicycle moves forward per complete revolution of the wheel, and explain why it equals the circumference.

A laboratory centrifuge rotates with a frequency of $f = 50\ \text{Hz}$ and the samples describe circles of radius $0.1\ \text{m}$.

1. a) Calculate the angular velocity $\omega = 2\pi \times f$.
2. b) Calculate the linear velocity of the samples.
3. c) Calculate their normal acceleration and express it in standard form.

A driver travels at $90\ \text{km/h}$, takes $0.8\ \text{s}$ to react to an obstacle and then brakes at $a = -5\ \text{m/s}^2$. The diagram shows the two stages of stopping.

1. a) Express the speed in $\text{m/s}$ and calculate the distance travelled during the reaction time.
2. b) Calculate the braking time and braking distance.
3. c) Calculate the total stopping distance, which is the sum of the previous two.

A car starts from rest with constant acceleration $a = 2\ \text{m/s}^2$ just as a motorbike passes alongside it with a constant speed of $12\ \text{m/s}$ in the same direction. The $x$-$t$ graphs for both vehicles are shown in the figure.

1. a) Write the position equations for both vehicles, taking the overtaking point as the origin.
2. b) Calculate the time at which the car catches up with the motorbike.
3. c) At what position does this happen, and what speed is the car travelling at that moment?
4. d) Explain, based on how the car's velocity varies with time, why it ends up catching up with it even though it falls behind at first.

A ball is thrown vertically upwards at $v_0 = 10\ \text{m/s}$ from a balcony $15\ \text{m}$ above the ground. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the maximum height it reaches, measured from the ground.
2. b) Set up the position equation taking the ground as the origin and calculate the total flight time until it reaches the ground.
3. c) Calculate the velocity with which it hits the ground.
4. d) Compare this flight time with the time it would take if it were simply dropped from the balcony, and explain the difference.

A car travels at $108\ \text{km/h}$.

1. a) Express this speed in $\text{m/s}$.
2. b) Calculate the braking distance on a dry road surface, with $a = -6\ \text{m/s}^2$.
3. c) Calculate the braking distance on a wet road surface, with $a = -3\ \text{m/s}^2$.
4. d) If the driver's reaction time is $0.5\ \text{s}$, calculate the total stopping distance in both cases and explain why it is advisable to increase the safe distance when it rains.

On a rotating platform at an amusement park rotating at $\omega = 0.5\ \text{rad/s}$, a child is sitting $2\ \text{m}$ from the axis and an adult at $8\ \text{m}$.

1. a) Calculate the linear velocity of each person.
2. b) Calculate the normal acceleration of each person.
3. c) Reason, using the expression $a_n = \omega^2 \times r$, which of the two experiences a greater acceleration and what this implies about the grip needed to avoid sliding outwards.

The drum of a washing machine has a radius of $0.25\ \text{m}$ and, during the spin cycle, rotates at $600$ revolutions per minute, that is, $10$ revolutions per second.

1. a) Express the rotation frequency in $\text{Hz}$ and calculate the angular velocity.
2. b) Calculate the linear velocity of a button stuck to the drum.
3. c) Calculate its normal acceleration and compare it with $g = 9.8\ \text{m/s}^2$: approximately how many times greater is it?
4. d) Explain how this result helps to understand why the spin cycle extracts water from clothes.

The minute hand of a station clock is $1.5\ \text{m}$ long and takes one hour to complete one full revolution.

1. a) Calculate its angular velocity in $\text{rad/s}$, in standard form.
2. b) Calculate the linear velocity of the tip of the hand.
3. c) Calculate the normal acceleration of the tip and express it in standard form.
4. d) Compare the angular velocity of the minute hand with that of the second hand, which takes $60\ \text{s}$ per revolution: how many times greater is that of the second hand?

A hot-air balloon ascends at a constant velocity of $8\ \text{m/s}$. When it is at a height of $48\ \text{m}$, a ballast bag becomes detached. Take $g \approx 10\ \text{m/s}^2$. Look at the diagram of the situation:

1. a) Explain why the bag, upon release, continues to rise for a few moments instead of falling immediately.
2. b) Set up the position equation of the bag taking the ground as the origin, and calculate the time it takes to reach the ground.
3. c) Calculate the velocity with which it hits the ground.
4. d) Calculate the height of the balloon at the moment the bag touches the ground.

In the "Rotor" ride, people lean against the inside wall of a cylinder with a radius of $4\ \text{m}$ that rotates until the floor is dropped. It is desired that the normal acceleration of the passengers is $2.5$ times $g$; take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the linear speed the wall must have.
2. b) Calculate the angular velocity $\omega$.
3. c) Calculate the period of rotation.
4. d) Explain why people do not fall when the floor drops away, mentioning which force provides the normal acceleration and which force balances the weight.