# An object with a mass of #4# #kg# is acted on by two forces. The first is #F_1=<-4# #N, 5# #N># and the second is #F_2 = <2# #N, -8# #N>#. What is the object's rate and direction of acceleration?

##### 1 Answer

The object is accelerating at

#### Explanation:

The net force,

#vec(F_"net") = vec(F_1) + vec(F_2)#

#= < -4 "N", 5 "N" > + < 2 "N", -8 "N" >#

#= < -2 "N", -3 "N" >#

Recall Newton's

#vec(F_"net") = m vec(a)#

where

Substituting the values in, the resulting equation is

#< -2 "N", -3 "N" > = (4 "kg") * < a_"x" , a_"y" >#

Solving for

#a_"x" = frac{-2 "N"}{4 "kg"} = -0.5 "m/s"^2#

#a_"y" = frac{-3 "N"}{4 "kg"} = -0.75 "m/s"^2#

The object's acceleration is found to be

#vec(a) = < -0.5 "m/s"^2, -0.75 "m/s"^2 >#

Its magnitude is

#||vec(a)|| = sqrt((-0.5 "m/s"^2)^2 + (-0.75 "m/s"^2)^2)#

#= 0.90 "m/s"^2#

Its direction is

#tan^{-1}(abs(frac{-0.75 "m/s"^2}{-0.5 "m/s"^2})) = 56.3^"o"#

The angle lies in the third quadrant since both the