A Bullet from a Moving Van

solved RelativeVelocity

Q. A police van doing 30 km/h $v_{v} = 30$ km/h fires a bullet at 150 m/s $v_{bv} = 150$ m/s, relative to the van, at a car ahead doing 192 km/h $v_{c} = 192$ km/h the same way. At what speed does the bullet hit the car, in m/s?

Inputs

van speed

v_{v}

30 km/h

car speed

v_{c}

192 km/h

muzzle speed

v_{bv}

150.0 m/s

Outputs

speed

v_{bc}

How it's solved

Solved with RelativeVelocity

Given

$v_{v} = 30$ km/h
$v_{c} = 192$ km/h
$v_{bv} = 150$ m/s

Find $v_{bc}$ .

Working

$v_{v} = 30$ km/h $= 8.33333333$ m/s
$v_{c} = 192$ km/h $= 53.3333333$ m/s
changing frames(bullet, van, ground) $v_{bv} = v_{b} - v_{v}$
$150 = v_{b} - 8.33333333$
$v_{b} = 158.333333$ m/s
changing frames(bullet, car, ground) $v_{bc} = v_{b} - v_{c}$
$v_{bc} = 158.333333 - 53.3333333$
$v_{bc} = 105.0 m/s$

Conclusion

What does the damage is the bullet's speed as the car sees it: the van's speed adds to the bullet's, and the car's running away takes from it.

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 :a_bullet_from_a_moving_van do4  kind :numerical5  describes "A police van doing %{van_speed} fires a bullet at %{muzzle_speed}, relative to the van, " \6          "at a car ahead doing %{car_speed} the same way. %{speed}, in m/s?"7  given van_speed: "30 km/h", car_speed: "192 km/h", muzzle_speed: "150 m/s"8  asks speed: "At what speed does the bullet hit the car"9end

solution.rb

1require_relative "../questions/a_bullet_from_a_moving_van"2require_relative "../science/motion_in_a_straight_line/relative_velocity"34solve :a_bullet_from_a_moving_van, using: RelativeVelocity do5  objects :bullet, :van, :car6  take van_speed: van.velocity, car_speed: car.velocity, muzzle_speed: bullet.velocity(from: van)7  speed = bullet.velocity(from: car)8  answer speed:, in: "m/s"9  conclusion "What does the damage is the bullet's speed as the car sees it: the van's speed adds " \10             "to the bullet's, and the car's running away takes from it."11end

playground.rb

1require_relative "../solutions/a_bullet_from_a_moving_van"23play_with :a_bullet_from_a_moving_van do4  title "A Bullet from a Moving Van"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
velocity	$v$	m/s	—	Acceleration
change of velocity	$Δ v$	m/s	—	Acceleration
average acceleration	$a_{av}$	m/s²	—	Acceleration
starting velocity	$v_{0}$	m/s	—	Acceleration
final velocity	$v$	m/s	—	Acceleration
velocity of ground from bullet	$v_{gb}$	m/s	—
velocity of ground from van	$v_{gv}$	m/s	—
velocity of bullet from van	$v_{bv}$	m/s	—
velocity of ground from car	$v_{gc}$	m/s	—
velocity of bullet from car	$v_{bc}$	m/s	—
velocity of van from bullet	$v_{vb}$	m/s	—
velocity of van from car	$v_{vc}$	m/s	—
velocity of car from bullet	$v_{cb}$	m/s	—
velocity of car from van	$v_{cv}$	m/s	—
velocity of bullet from ground	$v_{b}$	m/s	—
velocity of van from ground	$v_{v}$	m/s	—
velocity of car from ground	$v_{c}$	m/s	—

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

starting position

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

final position

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

starting velocity

v \equiv v_{i} (end of motion)

final velocity

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

v_{ab} (t) = v_{ac} (t) - v_{bc} (t)

changing frames

Δ x = x - x_{0}

change of position

v_{av} = \frac{Δ x}{Δ t}

displacement a time

c_{av} = \frac{d}{Δ t}

path a time

Δ v = v - v_{0}

velocity gained

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time

v_{gb} = v_{gv} - v_{bv}

changing frames(ground, bullet, van)

v_{gb} = v_{gc} - v_{bc}

changing frames(ground, bullet, car)

v_{gv} = v_{gb} - v_{vb}

changing frames(ground, van, bullet)

v_{gv} = v_{gc} - v_{vc}

changing frames(ground, van, car)

v_{gc} = v_{gb} - v_{cb}

changing frames(ground, car, bullet)

v_{gc} = v_{gv} - v_{cv}

changing frames(ground, car, van)

v_{b} = v_{bv} - v_{gv}

changing frames(bullet, ground, van)

v_{b} = v_{bc} - v_{gc}

changing frames(bullet, ground, car)

v_{bv} = v_{b} - v_{v}

changing frames(bullet, van, ground)

v_{bv} = v_{bc} - v_{vc}

changing frames(bullet, van, car)

v_{bc} = v_{b} - v_{c}

changing frames(bullet, car, ground)

v_{bc} = v_{bv} - v_{cv}

changing frames(bullet, car, van)

v_{v} = v_{vb} - v_{gb}

changing frames(van, ground, bullet)

v_{v} = v_{vc} - v_{gc}

changing frames(van, ground, car)

v_{vb} = v_{v} - v_{b}

changing frames(van, bullet, ground)

v_{vb} = v_{vc} - v_{bc}

changing frames(van, bullet, car)

v_{vc} = v_{v} - v_{c}

changing frames(van, car, ground)

v_{vc} = v_{vb} - v_{cb}

changing frames(van, car, bullet)

v_{c} = v_{cb} - v_{gb}

changing frames(car, ground, bullet)

v_{c} = v_{cv} - v_{gv}

changing frames(car, ground, van)

v_{cb} = v_{c} - v_{b}

changing frames(car, bullet, ground)

v_{cb} = v_{cv} - v_{bv}

changing frames(car, bullet, van)

v_{cv} = v_{c} - v_{v}

changing frames(car, van, ground)

v_{cv} = v_{cb} - v_{vb}

changing frames(car, van, bullet)