Use `dplyr` to sum all of the lagging values

Question

I want to sum all of the lagging values in a group.

Using cars for example data;

#This returns what I want
for (i in 1:nrow(cars)) {
  First<-which(cars$speed==cars[i,"speed"])[1]
  cars[i,"DistSum"]<-sum(cars[First:i,"dist"], na.rm = T)
}

head(cars)

However, I do not want to use a for loop for this, and I would like the following to return what I want. If I could get lag to return all of the previous values in a group that should do the trick.

#dplyr version that does not work
cars <- (cars %>% 
                 dplyr::group_by(speed) %>%
                 dplyr::mutate(DistSum = sum(lag(dist))) %>%
                   ungroup())

Another way of thinking of it might be that I need sum(lag(dist, n = ALL)).