You already know how to write position, velocity and acceleration as vectors. This section applies those tools to the most common types of motion: those that follow a straight line (with or without acceleration) and those that trace a circle at a constant rate. They are the building blocks from which, in the next section, more complex motions such as projectile motion are built.
Uniform motion and uniformly accelerated motion in vector notation
Imagine a high-speed train covering equal distances every second, compared with a car pulling away from traffic lights and steadily gaining speed: how can we express mathematically these two ways of moving in a straight line using vectors?
Move the sliders for $x_0$, $v$ (initial velocity) and $a$. With $a=0$ you recover constant-velocity motion: a straight line on the $x(t)$ graph. With $a \neq 0$, the $x(t)$ graph curves and $v(t)$ stops being horizontal: both equations are the same as ever, only now $x_0$, $v_0$ and $a$ are the components of the vectors $\vec{r}_0$, $\vec{v}_0$ and $\vec{a}$ along the axis of the path.
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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…
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 $x_0 = 100\ \text{m}$.
A cyclist moves in a straight line with uniform acceleration (constant acceleration), with $x_0 = 0$, $v_0 = 4\ \text{m/s}$ and $a = 1\ \text{m/s}^2$. a) Write its…
A cyclist moves in a straight line with uniform acceleration (constant acceleration), with $x_0 = 0$, $v_0 = 4\ \text{m/s}$ and $a = 1\ \text{m/s}^2$.
a) Write its position equation $x(t)$.
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 bus traveling at $72\ \text{km/h}$ brakes uniformly at $a = -2\ \text{m/s}^2$ until it comes to a stop. a) Express its initial velocity in $\text{m/s}$. b) Calculate…
A bus traveling at $72\ \text{km/h}$ brakes uniformly at $a = -2\ \text{m/s}^2$ until it comes to a stop.
If you throw a ball vertically upwards, you see it lose speed until it stops in mid-air and then falls back down faster and faster: how does a single equation describe both the rise and the fall under the invisible action of gravity?
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A coin is dropped from the top of a $20\ \text{m}$ tower. Take $g \approx 10\ \text{m/s}^2$. a) Calculate the time it takes to reach the ground. b) Calculate the…
A coin is dropped from the top of a $20\ \text{m}$ tower. Take $g \approx 10\ \text{m/s}^2$.
a) Calculate the time it takes to reach the ground.
A stone is thrown vertically downward 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: a) Set up the…
A stone is thrown vertically downward 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:
a) Set up the position equation taking the water surface as the origin and calculate the time it takes to reach the water.
A juggler throws a ball vertically upward at $v_0 = 30\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$. Look at the launch diagram: a) Calculate the time it takes to…
A juggler throws a ball vertically upward at $v_0 = 30\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$. Look at the launch diagram:
a) Calculate the time it takes to reach maximum height.
b) Calculate this maximum height.
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.
Uniform circular motion: period, frequency and velocities
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The blades of a ceiling fan take $0.5\ \text{s}$ to complete one full revolution. a) What are the period $T$ and the frequency $f$ of the rotation? b) Calculate the…
The blades of a ceiling fan take $0.5\ \text{s}$ to complete one full revolution.
a) What are the period $T$ and the frequency $f$ of the rotation?
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…
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.
a) Calculate the linear velocity $v = \omega \times r$ of a car located on the rim.
b) Explain why a car located closer to the axle would move more slowly, even though the angular velocity is the same.
A merry-go-round with a radius of $4\ \text{m}$ makes one revolution every $8\ \text{s}$. a) Calculate its angular velocity $\omega$ in $\text{rad/s}$, in terms of…
A merry-go-round with a radius of $4\ \text{m}$ makes one revolution every $8\ \text{s}$.
a) Calculate its angular velocity $\omega$ in $\text{rad/s}$, in terms of $\pi$.
b) Calculate the linear velocity of a child riding on the edge.
c) Calculate their normal acceleration. Express the final results to one decimal place as well.
When riding on a merry-go-round that turns at a steady rate, a speedometer would show a constant speed, but you feel your body tending to continue in a straight line and needing a force to avoid being flung off: if the value of the speed does not change, why do you experience an acceleration?
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A skater takes a curve of radius $8\ \text{m}$ at $4\ \text{m/s}$. The diagram shows the curve and the velocity vector. a) Calculate their normal acceleration…
A skater takes a curve of radius $8\ \text{m}$ at $4\ \text{m/s}$. The diagram shows the curve and the velocity vector.
a) Calculate their normal acceleration $a_n = \frac{v^2}{r}$.
A laboratory centrifuge rotates with a frequency of $f = 50\ \text{Hz}$ and the samples move in circles of radius $0.1\ \text{m}$. a) Calculate the angular velocity…
A laboratory centrifuge rotates with a frequency of $f = 50\ \text{Hz}$ and the samples move in circles of radius $0.1\ \text{m}$.
a) Calculate the angular velocity $\omega = 2\pi \times f$.
b) Calculate the linear velocity of the samples.
c) Calculate their normal acceleration and express it in scientific notation.
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 $8\ \text{m}$. a)…
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 $8\ \text{m}$.
a) Calculate the linear velocity of each person.
b) Calculate the normal acceleration of each person.
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 outward.
Accelerated circular motion and the components of acceleration
If $\alpha$ is constant, the motion is uniformly accelerated circular motion: $\omega=\omega_0+\alpha t$ and $\theta=\theta_0+\omega_0t+\tfrac12\alpha t^2$. Radians are used; $\vec a$ is not constant because its direction changes, even though $\alpha$ is constant.
On a bend of radius $50\ \text{m}$ taken at $10\ \text{m/s}$, $a_n=2\ \text{m/s}^2$. Even though the speedometer does not change, a net force towards the centre is needed; "centripetal force" is the radial resultant, not an additional force on top of the weight, the normal reaction or friction.
The diagram shows how the main ideas are related:
Let's practice! 8 medium exercises in this section