Mann died in the street shortly after that July 11 shooting. Mann's family has accused officers Randy Lozoya and John Tennis of escalating the situation and showing no regard for his life, even before they jumped out of the car, fired 18 bullets and shot Mann 14 times.
His family says the video, the audio of which was enhanced by the Sacramento Bee and published on Friday, is damning.
"They are officers that shouldn't be in uniform," Robert Mann, Joseph Mann's brother, told the newspaper. "If this is their state of mind when they go to work, this doesn't serve anyone well."
Lozoya and Tennis have been placed on desk duty. Mann's family has filed a federal lawsuit and a claim against the city, which they plan to update to include the attempts to hit Mann with the car, the newspaper reported.
Police released the dash-cam video two weeks ago, after resisting for two months following the fatal confrontation.
On July 11, they were called to a brightly lit street in California's capital by neighbours who reported a man standing in the street, waving a knife.
That was Joseph Mann, who was in the midst of a mental health crisis, according to court documents filed by his family. They say he was "doing karate moves and zigzagging back and forth across the street as he tried to walk away from the Officers."
Mann also had methamphetamine in his system, according to police. And he wasn't cooperating with the officers who arrived.
A lawsuit filed by Mann's family said he was "displaying obvious signs of mental distress." It faults the department for not trying to de-escalate the situation.
"Inexplicably, the Officers failed to contact any properly trained mental health counselors or make any attempt to use less than lethal force and ignored the established police protocols to make attempts to de-escalate the situation," the lawsuit said.
"Instead, Officers left their (positions) of safety, cornered Mr. Mann against a (fence/wall) and opened fire on him."
The shooting happened as officers across the nation have been embroiled in a debate about whether they are too quick to use lethal force, especially against minorities.
Police have killed 719 people so far this year, according to a Washington Post examination of fatal force. Like Mann, approximately 1 in 4 of those killed were reported to have some form of mental illness.
A few days before Mann was shot to death, a gunman opened fire on police officers at a Black Lives Matter protest in Dallas, killing five and increasing tensions between police and the communities they serve.
Mann's shooting sparked protests in Sacramento as residents demanded the release of the dash-cam videos.
For months, police claimed the videos were not public record because they were part of the investigative file. But two weeks ago, amid mounting public and political pressure, officials relented, releasing video from three dashboard cameras and one from a nearby business.
The most incendiary one has been the video shot through the windshield of Lozoya and Tennis's police cruiser.
It shows the series of close calls as the officers tried to hit Mann. After the second attempt, the vehicle is parked and two officers can be seen running in Mann's direction. Then, shots are fired.
