The velocity of a particle is zero at t=0

Options

A. The accelerationat t=0 must be zero
B. the acceleration at t=0 maybet zero
C. If the acceleratin is zero from t=0 to t=10 s, the speed is also zero in this interval.
D. If the speed is zer4o from t=0 to t=10 s the acceleration is zero in this interval.

Answer

To solve the question regarding the velocity of a particle being zero at \( t = 0 \), we need to analyze the implications of this statement on the acceleration and speed of the particle over time. Here’s a step-by-step breakdown of the reasoning:

### Step 1: Understand the Initial Condition
Given that the velocity of the particle at \( t = 0 \) is zero, we can denote this as:
\[ v(0) = 0 \quad \text{or} \quad u = 0 \]
where \( u \) is the initial velocity.

### Step 2: Analyze the Acceleration
The question presents various statements about acceleration at \( t = 0 \). We need to determine if the acceleration must be zero, may be zero, or if it affects the speed over time.

1. **Acceleration at \( t = 0 \) must be zero**: This statement is **incorrect**. Just because the velocity is zero at \( t = 0 \) does not imply that the acceleration must also be zero. A particle can have an initial velocity of zero and still experience acceleration (for example, if it starts from rest and begins to accelerate).

2. **Acceleration at \( t = 0 \) may be zero**: This statement is **correct**. The acceleration can indeed be zero, but it is not a necessity. The particle could be at rest or could start accelerating immediately.

### Step 3: Consider the Speed Over Time
Next, we need to analyze the implications of acceleration on speed over a time interval.

1. **If acceleration is zero from \( t = 0 \) to \( t = 10 \) seconds**: If the acceleration remains zero for this interval, then the speed will not change. Since the initial speed is zero and there is no acceleration to change it, the speed will remain zero throughout this interval. Thus, the speed at \( t = 10 \) seconds is also zero.

2. **If speed is zero from \( t = 0 \) to \( t = 10 \) seconds**: This statement is also **correct**. If the speed is zero at \( t = 0 \) and there is no acceleration to change it, then the speed will remain zero at \( t = 10 \) seconds.

### Conclusion
Based on the analysis:
- The initial velocity being zero does not imply that the acceleration must be zero.
- The acceleration may be zero.
- If acceleration is zero, the speed remains zero throughout the interval.

### Summary of Correct Statements
- Acceleration at \( t = 0 \) must be zero: **False**
- Acceleration at \( t = 0 \) may be zero: **True**
- If acceleration is zero from \( t = 0 \) to \( t = 10 \) seconds, the speed is also zero: **True**
- If speed is zero from \( t = 0 \) to \( t = 10 \) seconds, the speed remains zero: **True**