Question

An object travels North at `8 m/s` for `3 s` and then travels South at `7 m/s` for `8 s`. What are the object's average speed and velocity?

Answer 1

Average Speed

`bar(v)~~7.27color(white)(l)"m"*"s"^(-1)`

Average Velocity

`bar(sf(v))~~5.54color(white)(l)"m"*"s"^(-1)`

Explanation:

`"Speed"` equals distance over time whereas
`"Velocity"` equals displacement over time.

Total distance travelled- which is independent of the direction of motion- in `3+8=11color(white)(l)"seconds"`

`Delta s=s_1+s_2=v_1*t_1+v_2*t_2=8*3+7*8=80color(white)(l)"m"`

Average speed

`bar(v)=(Delta s)/(Delta t)=(80color(white)(l)"m")/(11color(white)(l)"s")~~7.27color(white)(l)"m"*"s"^(-1)`

The two components of the final displacement, `sf(x)_1` and `sf(x)_2`, are normal (a.k.a., perpendicular ) to each other.

Hence, directly apply the Pythagorean theorem to find the
displacement from the initial position after `11color(white)(l)"seconds"`

`Delta sf(x)=sqrt(sf(x)_1^2+sf(x)_2^2)=sqrt((sf(v)_1*t_1)^2+(sf(v)_2*t_2)^2)`

`=sqrt((8*3)^2+(7*8)^2)=8sqrt(58)color(white)(l)"m"`

Average velocity

`bar(sf(v))=(Delta sf(x))/(Delta t)=(8sqrt(58)color(white)(l)"m")/(11color(white)(l)"s")~~5.54color(white)(l)"m"*"s"^(-1)`