Question

A body is moving under the action of two forces ${\vec F_1} = 2\hat i - 5\hat j$ ; ${\vec F_2} = 3\hat i - 4\hat j$ . Its velocity will become uniform under a third force ${\vec F_3}$ given by
A. $5\hat i - \hat j$
B. $- 5\hat i - \hat j$
C. $5\hat i + \hat j$
D. $- 5\hat i + 9\hat j$

Answer 1

Uniform velocity- When the body covers equal distances in equal intervals of time in a particular direction it is said to be moving with a uniform velocity. There is no change in speed or direction observed in this kind of motion. Example – Movement of hands of a clock. Rotation of Earth about its axis.

Complete step by step answer:
For uniform velocity, there is no change in magnitude or direction in the motion of the body. Hence there is no acceleration. Acceleration is the rate at which velocity of a body changes with time. In the case of uniform velocity acceleration is zero.
Newton’s Second law of Motion states that rate of change of momentum of a body is directly proportional to the force applied on it.
We know,
$F = ma$ where $F$ is the force applied on the body, $m$ is the mass of the body, $a$ is the acceleration of the body.
If the acceleration of the body is zero, the force acting on the body also becomes zero.
Therefore, the resultant force acting on a body moving with uniform speed will be zero.
${\vec F_1} + {\vec F_2} + {\vec F_3} = 0$
$\Rightarrow (2\hat i - 5\hat j) + (3\hat i - 4\hat j) + {\vec F_3} = 0$
$\Rightarrow {F_3} = ( - 2 - 3)\hat i + (5 + 4)\hat j = - 5\hat i + 9\hat j$
Option D. $- 5\hat i + 9\hat j$ is correct.

Note:
Velocity, Acceleration and Force are vector quantities having both magnitude and direction. Vectors along the direction of the $x$ axis are denoted with $\hat i$ and vectors along the direction of the $y$ axis are denoted with $\hat j$ and vectors along the direction of $z$ are denoted with $\hat k$ .