Final answer:
American military leaders aimed to showcase US resolve and firepower through Operation Rolling Thunder, to support South Vietnam in defeating the North, and to push the North Vietnamese government to surrender by bombing military targets, though civilian casualties were a significant issue. The expectation was that the North's will to fight would be broken by heavy losses caused by superior US firepower.
Step-by-step explanation:
The American military leaders had several objectives in North Vietnam with campaigns such as Operation Rolling Thunder. Primarily, the campaign was intended to demonstrate US resolve and firepower, to support the South Vietnamese in defeating the North, and to coerce the North Vietnamese government into surrendering. This was to be achieved by bombing military targets across North Vietnam, but distinguishing military targets from civilians was a complex issue. Civilian casualties were often a tragic consequence of these efforts.
Moreover, the strategy under General William Westmoreland focused on leveraging superior U.S. firepower to inflict heavy losses on the North Vietnamese, which was supposed to break their will to fight. However, the indiscriminate bombing failed to discriminate between military and civilian targets properly. U.S. military leaders also hoped to cut off the supply lines to the Vietcong and North Vietnamese and sustain pressure that would lead to a loss of morale and eventual capitulation.
Despite the humanitarian aid provided by the US to counteract the negative effects of their military strategy and gain the support of Vietnamese villagers, the strategy was not as effective as hoped. The war on the countryside and the free-fire zones did not lead to the desired strategic outcomes, and the 'hearts and minds' campaign often ended up pushing more Vietnamese towards supporting the North Vietnamese cause.