Question

A small boat is crossing a lake on a windy day. During some interval of time, the boat undergoes the given displacement Δ → r . Δr→. Δ → r = ( 3.11 m ) ^ ı + ( 2.69 m ) ^ ȷ Δr→=(3.11 m)ı^+(2.69 m)ȷ^ During the same interval of time, the wind exerts the given constant force → F F→ on the boat. → F = ( 263 N ) ^ ı − ( 131 N ) ^ ȷ F→=(263 N)ı^−(131 N)ȷ^ What is the total work done on the boat by the wind over this period of time?

Answer 1

The workdone is
`W= 465.54J`

Step-by-step explanation:

from the question we are told that

The displacement of the boat is
`\Delta \= r = [3.11 \ m]\ \r i \ + [2.69\ m] \ \r j`

The force exerted on the boat is
`\= F = [263 N]\ \r i - [131\ N] \ \r j`

The workdone is mathematically represented as

`W = \= F \cdot \Delta \= r`

`= [ ( [263 N]\ \r i - [131\ N] \ \r j)( [3.11 \ m]\ \r i \ + [2.69\ m] \ \r j)]`

`=`
`( [3.11 \ m]\ \r i ) ([263 N]\ \r i) +( [2.69\ m] \ \r j)(([263 N]\ \r i)`

`- ( [3.11 \ m]\ \r i ) ([131\ N] \ \r j) - ([263 N]\ \r i) ([131\ N] \ \r j)`

`W = 817.93-352.39`

`W= 465.54J`

Note: Generally

`(i * i) =1\\(j * j) = 1\\(i *j) =0\\(j*i)=0`