The kinetic energy of a particle continuously increases with time

Options

A. the resultant force on the particle must be parallel to the velocity at all instants.
B. the resultant force on the particle must be at an angle less than `90^0` all the time
C. Its height above the ground level must continuously decrease
D. the magnitude of its linear momentum is increasing continuously.

Answer

To solve the problem regarding the kinetic energy of a particle that continuously increases with time, we will analyze the implications of this statement step by step.

### Step 1: Understand Kinetic Energy
Kinetic energy (KE) of a particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 2: Implication of Increasing Kinetic Energy
If the kinetic energy of the particle is continuously increasing, it implies that the velocity of the particle must also be increasing, assuming the mass remains constant. This means:
\[ \frac{d(KE)}{dt} > 0 \]

### Step 3: Relate Work and Kinetic Energy
According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy:
\[ W = \Delta KE \]
Since KE is increasing, the work done must be positive:
\[ W > 0 \]

### Step 4: Analyze the Resultant Force
The work done on the particle is also related to the force acting on it. The work done by a force is given by:
\[ W = F \cdot d \]
where \( F \) is the force and \( d \) is the displacement. For the work to be positive, the force must have a component in the direction of the displacement. This means that the resultant force must be at an angle less than 90 degrees with respect to the direction of velocity.

### Step 5: Consider Linear Momentum
The linear momentum \( p \) of the particle is given by:
\[ p = mv \]
If the kinetic energy is increasing, the velocity is increasing, which means the linear momentum must also be increasing:
\[ \frac{dp}{dt} = m \frac{dv}{dt} > 0 \]

### Step 6: Conclusion
From the analysis, we can conclude:
1. The resultant force on the particle must be at an angle less than 90 degrees to the velocity to ensure positive work is done.
2. The magnitude of the linear momentum is continuously increasing as the kinetic energy increases.

### Final Answer
Based on the analysis, the correct statements regarding the particle with continuously increasing kinetic energy are:
- The resultant force on the particle must be at an angle less than 90 degrees to the velocity.
- The magnitude of the linear momentum is continuously increasing.