The linear momentum of a sphere of mass of 2.0 kg varies with time t as P = 3t$^2$ + 4, where P and t are in S.I. units. It follows that the sphere moving with a :

Question

Accepted Answer

The problem involves understanding the relationship between linear momentum and acceleration for a sphere whose momentum varies with time. Let's analyze why Option 2, "Variable acceleration," is the correct answer.

### Explanation for Option 2: Variable acceleration

1. **Momentum and Force Relationship**: The linear momentum $ P $ of the sphere is given by the equation $ P = 3t^2 + 4 $. In physics, the force $ F $ acting on an object is the time derivative of its momentum, i.e., $ F = \frac{dP}{dt} $.

2. **Calculating the Force**: Differentiating the given momentum equation with respect to time $ t $, we get:
   $$
   F = \frac{dP}{dt} = \frac{d}{dt}(3t^2 + 4) = 6t
   $$
   This shows that the force acting on the sphere is $ 6t $, which is clearly dependent on time and not constant.

3. **Acceleration and Force Relationship**: According to Newton's second law, $ F = ma $, where $ m $ is the mass and $ a $ is the acceleration. Given the mass $ m = 2.0 \, \text{kg} $, we can find the acceleration:
   $$
   a = \frac{F}{m} = \frac{6t}{2} = 3t
   $$
   The acceleration $ a = 3t $ is a function of time, indicating that the acceleration is variable.

Thus, the sphere is moving with a variable acceleration, which supports Option 2 as the correct answer.

### Explanation for why other options are incorrect:

1. **Option 1: Constant acceleration**
   - As derived above, the acceleration $ a = 3t $ is not constant; it changes with time. Therefore, the sphere does not have constant acceleration.

2. **Option 3: Constant speed**
   - Constant speed implies zero acceleration. Since we have established that the acceleration is $ 3t $, which is not zero, the sphere cannot be moving at a constant speed.

3. **Option 4: Variable speed**
   - While the speed is indeed variable due to the variable acceleration, the question specifically asks about the nature of the acceleration. The correct focus is on the acceleration being variable, not just the speed.

In conclusion, the correct interpretation of the given momentum equation and its implications on force and acceleration confirms that the sphere is moving with a variable acceleration, making Option 2 the correct answer.