Question

A particle which is experiencing a force given by $\vec F = 3\hat i - 12\hat j$ undergoes a displacement of $\vec d = 4\hat i$. If the particle had a kinetic energy of $3{\text{J}}$ at the beginning of the displacement, find its kinetic energy at the end of the displacement.
A) ${\text{15J}}$
B) ${\text{10J}}$
C) ${\text{12J}}$
D) ${\text{9J}}$

Answer 1

Since the force vector and the displacement vector of the particle are given, the dot product of these two vectors will give the work done by the force acting on the particle. The work-energy theorem states that this work done will be equal to the change in the kinetic energy of the particle.

Formulas used:
The work done by a force acting on a body is given by, $W = F \cdot d$ where $F$ is the applied force and $d$ is the corresponding displacement of the object.
The change in kinetic energy of an object is given by, $\Delta K = {K_f} - {K_i}$ where ${K_f}$ and ${K_i}$ are the final kinetic energy and initial kinetic energy of the object respectively.

Complete step by step answer:
The force acting on the particle is represented as $\vec F = 3\hat i - 12\hat j$ and its corresponding displacement is represented as $\vec d = 4\hat i$.
The initial kinetic energy of the particle is given to be ${K_i} = 3{\text{J}}$ and its final kinetic energy ${K_f}$ has to be determined.
The work done by the force acting on the particle is given by,
$W = F \cdot d$ --------- (1)
where $F$ is the applied force and $d$ is the corresponding displacement of the object.
Substituting for $\vec F = 3\hat i - 12\hat j$ and $\vec d = 4\hat i$ in equation (1) we get,
$\Rightarrow W = \left( {3\hat i - 12\hat j} \right) \cdot \left( {4\hat i} \right) = 12{\text{J}}$
Thus the work done by the applied force is $W = 12{\text{J}}$.
According to the work-energy theorem, the change in kinetic energy will be equal to the work done by the force acting on the particle.
i.e., $\Delta K = W = 12{\text{J}}$
The change in kinetic energy of the particle is given by, $\Delta K = {K_f} - {K_i}$ -------- (2)
Substituting for $\Delta K = 12{\text{J}}$ and ${K_i} = 3{\text{J}}$ in the above equation we get,
$\Rightarrow 12 = {K_f} - 3$
$\Rightarrow {K_f} = 12 + 3 = 15{\text{J}}$

Thus the final kinetic energy of the particle will be ${K_f} = 15{\text{J}}$. Hence the correct option is A.

Note:
The dot product of two vectors is obtained by multiplying the x-component of each vector and then adding the result to the product of the y-components of each vector.
i.e., $\vec F \cdot \vec d = \left( {3\hat i - 12\hat j} \right) \cdot \left( {4\hat i} \right) = \left[ {\left( {3 \times 4} \right) + \left( { - 12 \times 0} \right)} \right] = 12$
Force and displacement are vectors while the work done is a scalar quantity. Hence the dot product is taken.