A car of mass m starts from rest and acquires a velocity along east v = vî (v>0) in two seconds. Assuming the car moves with uniform acceleration, the force exerted on the car is

The correct option is: mv/2, eastward and is due to the friction on the tyres exerted by the road.

Let us analyze the motion of the car step-by-step to find both the magnitude and direction of the force acting on it, as well as its origin:

1. Finding the acceleration of the car:
The car starts from rest, which means its initial velocity is:
$\overrightarrow{u}=0$
It acquires a final velocity along the east direction in a time interval of $t=2\text{seconds}$:
$\overrightarrow{v}=v\hat{i}$ (where $\hat{i}$ represents the unit vector pointing east)
Assuming the car moves with a uniform acceleration $\overrightarrow{a}$, we can use the first equation of motion:
$\overrightarrow{v}=\overrightarrow{u}+\overrightarrow{a}t$
Substituting the given values:
$v\hat{i}=0+\overrightarrow{a}\left(2\right)$
Solving for acceleration $\overrightarrow{a}$:
$\overrightarrow{a}=\frac{v}{2}\hat{i}$

2. Finding the net external force:
According to Newton's second law of motion, the net external force $\overrightarrow{F}$ acting on the car of mass $m$ is:
$\overrightarrow{F}=m\overrightarrow{a}$
Substituting the value of acceleration:
$\overrightarrow{F}=\frac{mv}{2}\hat{i}$
This shows that the net force has a magnitude of $mv/2$ and is directed eastward (along $\hat{i}$).

3. Determining the source of the force:
An internal force (like the force exerted by the engine inside the car) cannot change the velocity of the center of mass of the system. For a car to accelerate forward, it requires an external force.
When the engine rotates the wheels, the tyres push backwards against the road. By Newton's third law, the road exerts an equal and opposite force on the tyres in the forward direction. This external force is the force of static friction.
Therefore, the force of magnitude $mv/2$ acting eastward is exerted by the road on the tyres due to friction.