Question

An object of mass 4 Kg starts at the origin in xy-plane with initial velocityvector,( 0 )1, 1, 1v=<>. A force of magnitude 20 N acts directly upward from the xy-plane underneath the object, while gravitational force acts downward on the object. The magnitude of gravitational acceleration210 m/sg≈. Find, (i) the velocity()vtand the position function ()rtof this object, (ii) the time it lands on ground, and (iii) the coordinates of its landing.

Answer 1

1) we know

`F=ma\\a=F/m\\a=20/4=3m/s^2`

since the force is acting upward and the acceleration is in positive z-direction

`a(t)=5k`

we integrate it to get the velocity

`v(t)=5tk+C`

where C is integration constant

`Given\\v(0)=i-j`

using this to find C

`v(0)=5(0)k+C\\i-j=0+C\\C=i-j`

substitute back the value of C we get:

`v(t)=i-j+5tk`

integrate to get the position

`r(t)=ti-tj+(5t^2)/(2) k+C`

where C is zero because it starts from origin

speed is magnitude of velocity

`v(t)=i-j+5tk\\therefore\\\\v(t)=√(1^2+(-1)^2+(5t)^2) =√(2+25t^2)`