Forty Minutes Out: Velocity

solved Velocity

Q. A man walks on a straight road $v_{av} (to) = c_{av} (to)$ from his home $x (start) = 0$ km to a market 2.5 km $x (market) = 2.5$ km away with a speed of 5 km/h $c_{av} (to) = 5$ km/h. Finding the market closed, he instantly turns and walks back home $v_{av} (tu) = - c_{av} (tu)$ with a speed of 7.5 km/h $c_{av} (tu) = 7.5$ km/h. What is the magnitude of average velocity of the man over the interval of time 0 to 40 min $Δ t_{w} = 40$ min, in km/h?

Inputs

distance

x

2.5 km

speed there

c_{av}

5 km/h

speed back

c_{av}

7.5 km/h

interval

Δ t_{w}

40 min

Outputs

average velocity

v_{av}

How it's solved

Solved with Velocity

Given

$x (market) = 2.5$ km
$c_{av} (to) = 5$ km/h
$c_{av} (tu) = 7.5$ km/h
$Δ t_{w} = 40$ min
$x (start) = 0$ km from “from his home”
$v_{av} (to) = c_{av} (to)$ from “on a straight road”
$v_{av} (tu) = - c_{av} (tu)$ from “walks back home”

Find $v_{av} (w)$ .

Working

$x (market) = 2.5$ km $= 2500$ m
$c_{av} (to) = 5$ km/h $= 1.38888889$ m/s
$c_{av} (tu) = 7.5$ km/h $= 2.08333333$ m/s
$Δ t_{w} = 40$ min $= 2400$ s
$x (start) = 0$ km $= 0$ m
on a straight road $v_{av} (to) = c_{av} (to)$
$v_{av} (to) = 1.38888889$
$v_{av} (to) = 1.38888889$ m/s
change of position(to market) $Δ x (to) = x (market) - x (start)$
$Δ x (to) = 2500 - 0$
$Δ x (to) = 2500$ m
displacement a time(to market) $v_{av} (to) = \frac{Δ x (to)}{Δ t_{to}}$
$1.38888889 = \frac{2500}{Δ t_{to}}$
$Δ t_{to} = 1800$ s
time in parts(walking) $Δ t_{w} = Δ t_{to} + Δ t_{tu}$
$2400 = 1800 + Δ t_{tu}$
$Δ t_{tu} = 600$ s
walks back home $v_{av} (tu) = - c_{av} (tu)$
$v_{av} (tu) = - 2.08333333$
$v_{av} (tu) = −2.08333333$ m/s
displacement a time(turning back) $v_{av} (tu) = \frac{Δ x (tu)}{Δ t_{tu}}$
$(−2.08333333) = \frac{Δ x (tu)}{600}$
$Δ x (tu) = −1250$ m
change of position(turning back) $Δ x (tu) = x (later) - x (market)$
$(−1250) = x (later) - 2500$
$x (later) = 1250$ m
change of position(walking) $Δ x (w) = x (later) - x (start)$
$Δ x (w) = 1250 - 0$
$Δ x (w) = 1250$ m
displacement a time(walking) $v_{av} (w) = \frac{Δ x (w)}{Δ t_{w}}$
$v_{av} (w) = \frac{1250}{2400}$
$v_{av} (w) = 1.875$ km/h
$v_{av} (w) = 0.520833333$ m/s $= 1.875$ km/h
$v_{av} (w) \approx 1.875$ km/hto 4 significant figures

Conclusion

By the end of the interval he has walked part of the way back, so the average velocity counts only how far from home he is now, over the whole time.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :forty_minutes_out_velocity do4  kind :numerical5  describes "A man walks on a straight road from his home to a market %{distance} away with a speed of " \6            "%{speed_there}. Finding the market closed, he instantly turns and walks back home with a speed of " \7            "%{speed_back}. %{average_velocity} over the interval of time 0 to %{interval}, in km/h?"8  given distance: "2.5 km", speed_there: "5 km/h", speed_back: "7.5 km/h", interval: "40 min"9  asks average_velocity: "What is the magnitude of average velocity of the man"10end

solution.rb

1require_relative "../questions/forty_minutes_out_velocity"2require_relative "../science/motion_in_a_straight_line/velocity"34solve :forty_minutes_out_velocity, using: Velocity do5  objects :man6  instants :start, :market, :later7  man.goes(:to_market, during: start..market)8  man.goes(:turning_back, during: market..later)9  man.goes(:walking, during: start..later)10  take distance: man.position(at: market), speed_there: to_market.average_speed,11       speed_back: turning_back.average_speed, interval: walking.time_taken12  deduce_from "from his home", man.position(at: start) => "0 km"13  deduce_from "on a straight road", to_market.average_velocity => to_market.average_speed14  deduce_from "walks back home", turning_back.average_velocity => -turning_back.average_speed15  average_velocity = walking.average_velocity16  answer average_velocity:, in: "km/h", figures: 417  conclusion "By the end of the interval he has walked part of the way back, so the average velocity counts " \18             "only how far from home he is now, over the whole time."19end

playground.rb

1require_relative "../solutions/forty_minutes_out_velocity"23play_with :forty_minutes_out_velocity do4  title "Forty Minutes Out: Velocity"5end

The laws it drew on

quantity	written	units	within	from
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
displacement of man during to market	$Δ x (to)$	m	—
position of man at market	$x (market)$	m	—
position of man at start	$x (start)$	m	—
displacement of man during turning back	$Δ x (tu)$	m	—
position of man at later	$x (later)$	m	—
displacement of man during walking	$Δ x (w)$	m	—
average velocity of man during to market	$v_{av} (to)$	m/s	—
time taken of to market	$Δ t_{to}$	s	0 … 1e+05
average velocity of man during turning back	$v_{av} (tu)$	m/s	—
time taken of turning back	$Δ t_{tu}$	s	0 … 1e+05
average velocity of man during walking	$v_{av} (w)$	m/s	—
time taken of walking	$Δ t_{w}$	s	0 … 1e+05
average speed of man during to market	$c_{av} (to)$	m/s	—
path length of man during to market	$d (to)$	m	—
average speed of man during turning back	$c_{av} (tu)$	m/s	—
path length of man during turning back	$d (tu)$	m	—
average speed of man during walking	$c_{av} (w)$	m/s	—
path length of man during walking	$d (w)$	m	—

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

Δ x (to) = x (market) - x (start)

change of position(to market)

Δ x (tu) = x (later) - x (market)

change of position(turning back)

Δ x (w) = x (later) - x (start)

change of position(walking)

v_{av} (to) = \frac{Δ x (to)}{Δ t_{to}}

displacement a time(to market)

v_{av} (tu) = \frac{Δ x (tu)}{Δ t_{tu}}

displacement a time(turning back)

v_{av} (w) = \frac{Δ x (w)}{Δ t_{w}}

displacement a time(walking)

c_{av} (to) = \frac{d (to)}{Δ t_{to}}

path a time(to market)

c_{av} (tu) = \frac{d (tu)}{Δ t_{tu}}

path a time(turning back)

c_{av} (w) = \frac{d (w)}{Δ t_{w}}

path a time(walking)

d (w) = d (to) + d (tu)

path in parts(walking)

Δ t_{w} = Δ t_{to} + Δ t_{tu}

time in parts(walking)

v_{av} (to) = c_{av} (to)

on a straight road

v_{av} (tu) = - c_{av} (tu)

walks back home