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 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 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 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.

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.