Question

An object moves along x-axis such that its position varies with time according to the relation `x=50t-5t^(2)`
Here, x is in meters and time is in seconds. Find the displacement and distance travelled for the time interval t = 0 to t = 6 s.

Answer 1

Displacement = change in position
`rArr" Displacement "=x_(6)-x_(0)=[50xx6-5(6)^(2)]-[50xx0-5(0)^(2)]`
`=(300-180)-0`
`=120m`
Distance travelled for an object moving along a straight line can be equal to displacement when the particle does not turn back. To find, whether there is a turning point or not, find the point where velocity becomes zero
As `v=(dx)/(dt)" "rArr" "v=50-10t.`
Put `v=0," "0=50-10t rArr t=5 s.`
This value t = 5 s lies between the interval 0 - 6 s.
So, we have to describe the exact path.
The position at t = 0 is `x_(0)=50xx0-5(0)^(2)=0`
The position at t = 5 s is `x_(5)=50xx5-5(5)^(2)=125m`
The position at t = 6 s is `x_(6)=50xx6-5(6)^(2)=120m.`
So, the object moves as shown.

`"Distance travelled"=" 125 m (in forward direction)" + " 5 m (in backward direction)"`
`=130 m.`