A Bullet from a Moving Van

Q. A police van doing 30 km/h fires a bullet at 150 m/s, relative to the van, at a car ahead doing 192 km/h the same way. At what speed does the bullet hit the car, in m/s? expects 105.0 m/s

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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?

The playground

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}

The solution

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 Ruby

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

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

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

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)