Uniformly accelerated motion

Acceleration, velocity and displacement at constant acceleration, speeding up and braking, graphs.

Terms

Studied 0 of 5

Formulas

Acceleration

a_x = \dfrac{v_x - v_{0x}}{t}

Quantities
Symbol	Name	Unit
$a_x$	acceleration component	m/s²
$v_x$	final velocity	m/s
$v_{0x}$	initial velocity	m/s
$t$	time	s

Solved for another quantity:

v_x = v_{0x} + a_x\,t

t = \dfrac{v_x - v_{0x}}{a_x}

Velocity at time t

v_x = v_{0x} + a_x\,t

Quantities
Symbol	Name	Unit
$v_x$	velocity at time $t$	m/s
$v_{0x}$	initial velocity	m/s
$a_x$	acceleration component	m/s²
$t$	time	s

Solved for another quantity:

a_x = \dfrac{v_x - v_{0x}}{t}

t = \dfrac{v_x - v_{0x}}{a_x}

Note. Starting from rest: $v_{0x} = 0$ , so $v_x = a_x t$ . The body has stopped: $v_x = 0$ .

Displacement

s_x = v_{0x}\,t + \dfrac{a_x t^2}{2}

Quantities
Symbol	Name	Unit
$s_x$	displacement component	m
$v_{0x}$	initial velocity	m/s
$a_x$	acceleration component	m/s²
$t$	time	s

Note. From rest: $s_x = \dfrac{a_x t^2}{2}$ . Time enters squared: in twice the time the body covers four times the distance.

Coordinate

x = x_0 + v_{0x}\,t + \dfrac{a_x t^2}{2}

Quantities
Symbol	Name	Unit
$x$	coordinate at time $t$	m
$x_0$	initial coordinate	m
$v_{0x}$	initial velocity	m/s
$a_x$	acceleration component	m/s²
$t$	time	s

Solved for another quantity:

x = x_0 + s_x

Velocity and displacement

v_x^2 - v_{0x}^2 = 2\,a_x\,s_x

Quantities
Symbol	Name	Unit
$v_x$	final velocity	m/s
$v_{0x}$	initial velocity	m/s
$a_x$	acceleration component	m/s²
$s_x$	displacement component	m

Solved for another quantity:

s_x = \dfrac{v_x^2 - v_{0x}^2}{2 a_x}

Note. Handy when time is neither given nor required.

Displacement via mean velocity

s_x = \dfrac{v_{0x} + v_x}{2}\,t

Quantities
Symbol	Name	Unit
$s_x$	displacement component	m
$v_{0x}$	initial velocity	m/s
$v_x$	final velocity	m/s
$t$	time	s

Note. Valid only for uniformly accelerated motion: the mean velocity is half the sum of the initial and final ones.

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

Motion of the body

Drag the cart with the mouse to set its starting position. The dots mark the body's position every 1 s.

t

= 0 sPlayback speed

Graphs

The graphs are drawn as the body moves. Click a graph to move through the recording.

x(t)

s_x(t)

$s_x = x - x_0$ . It decreases when the body moves against the axis.

l(t)

Distance never decreases: after a turnaround it grows while $s_x$ shrinks.

v_x(t)

The shaded area is the displacement $s_x$ .

a_x(t)

t

0 s

s_x

0 m

l

0 m

Formulas with your numbers

v_x = v_{0x} + a_x t = 2 + 1\cdot 0 = 2\ \text{m/s}

x = x_0 + v_{0x}t + \dfrac{a_x t^2}{2} = -15 + 2\cdot 0 + \dfrac{1\cdot 0^{2}}{2} = -15\ \text{m}

Lab «Moving man»

Drag the man by hand — the x(t), v(t) and a(t) graphs draw themselves. Then set the motion with numbers or repeat a graph.

Open full screen →

How to read the graphs

$v_x(t)$ is a sloped line. The slope is the acceleration $a_x$ : up means $a_x > 0$ , down means $a_x < 0$ .
Where $v_x(t)$ crosses the time axis, the body stops and turns around.
$s_x(t)$ and $x(t)$ are parabolas. They open upwards for $a_x > 0$ and downwards for $a_x < 0$ .
The area under the $v_x(t)$ graph equals the displacement $s_x$ . Area below the time axis counts as negative.
$a_x(t)$ is a horizontal line: the acceleration is constant.

Loading…

Terms

Studied 0 of 5

Formulas

Acceleration

a_x = \dfrac{v_x - v_{0x}}{t}

Quantities
Symbol	Name	Unit
$a_x$	acceleration component	m/s²
$v_x$	final velocity	m/s
$v_{0x}$	initial velocity	m/s
$t$	time	s

Solved for another quantity:

v_x = v_{0x} + a_x\,t

t = \dfrac{v_x - v_{0x}}{a_x}

Velocity at time t

v_x = v_{0x} + a_x\,t

Quantities
Symbol	Name	Unit
$v_x$	velocity at time $t$	m/s
$v_{0x}$	initial velocity	m/s
$a_x$	acceleration component	m/s²
$t$	time	s

Solved for another quantity:

a_x = \dfrac{v_x - v_{0x}}{t}

t = \dfrac{v_x - v_{0x}}{a_x}

Note. Starting from rest: $v_{0x} = 0$ , so $v_x = a_x t$ . The body has stopped: $v_x = 0$ .

Displacement

s_x = v_{0x}\,t + \dfrac{a_x t^2}{2}

Quantities
Symbol	Name	Unit
$s_x$	displacement component	m
$v_{0x}$	initial velocity	m/s
$a_x$	acceleration component	m/s²
$t$	time	s

Note. From rest: $s_x = \dfrac{a_x t^2}{2}$ . Time enters squared: in twice the time the body covers four times the distance.

Coordinate

x = x_0 + v_{0x}\,t + \dfrac{a_x t^2}{2}

Quantities
Symbol	Name	Unit
$x$	coordinate at time $t$	m
$x_0$	initial coordinate	m
$v_{0x}$	initial velocity	m/s
$a_x$	acceleration component	m/s²
$t$	time	s

Solved for another quantity:

x = x_0 + s_x

Velocity and displacement

v_x^2 - v_{0x}^2 = 2\,a_x\,s_x

Quantities
Symbol	Name	Unit
$v_x$	final velocity	m/s
$v_{0x}$	initial velocity	m/s
$a_x$	acceleration component	m/s²
$s_x$	displacement component	m

Solved for another quantity:

s_x = \dfrac{v_x^2 - v_{0x}^2}{2 a_x}

Note. Handy when time is neither given nor required.

Displacement via mean velocity

s_x = \dfrac{v_{0x} + v_x}{2}\,t

Quantities
Symbol	Name	Unit
$s_x$	displacement component	m
$v_{0x}$	initial velocity	m/s
$v_x$	final velocity	m/s
$t$	time	s

Note. Valid only for uniformly accelerated motion: the mean velocity is half the sum of the initial and final ones.

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

Motion of the body

Drag the cart with the mouse to set its starting position. The dots mark the body's position every 1 s.

t

= 0 sPlayback speed

Graphs

The graphs are drawn as the body moves. Click a graph to move through the recording.

x(t)

s_x(t)

$s_x = x - x_0$ . It decreases when the body moves against the axis.

l(t)

Distance never decreases: after a turnaround it grows while $s_x$ shrinks.

v_x(t)

The shaded area is the displacement $s_x$ .

a_x(t)

t

0 s

s_x

0 m

l

0 m

Formulas with your numbers

v_x = v_{0x} + a_x t = 2 + 1\cdot 0 = 2\ \text{m/s}

x = x_0 + v_{0x}t + \dfrac{a_x t^2}{2} = -15 + 2\cdot 0 + \dfrac{1\cdot 0^{2}}{2} = -15\ \text{m}

How to read the graphs

v_x(t)

is a sloped line. The slope is the acceleration

a_x

: up means

a_x > 0

, down means

a_x < 0

Where

v_x(t)

crosses the time axis, the body stops and turns around.

s_x(t)

and

x(t)

are parabolas. They open upwards for

a_x > 0

and downwards for

a_x < 0

The area under the

v_x(t)

graph equals the displacement

s_x

. Area below the time axis counts as negative.

a_x(t)

is a horizontal line: the acceleration is constant.