To determine the acceleration of the particle at the given instant, we can use the information provided about the velocity and its rate of change with respect to displacement. The problem states that the velocity of the particle is decreasing at a rate of 8 m/s per meter of displacement. Let's break this down step by step.
Understanding the Relationship Between Velocity and Displacement
We know that the velocity \( v \) of the particle is given as 20 m/s, and the displacement \( s \) is 14 m. The rate at which the velocity is changing with respect to displacement is given as:
- Rate of change of velocity = -8 m/s per meter of displacement
The negative sign indicates that the velocity is decreasing. This can be expressed mathematically as:
Mathematical Representation
We can express the relationship as:
\( \frac{dv}{ds} = -8 \, \text{m/s/m} \)
Finding Acceleration
Acceleration \( a \) can be defined as the change in velocity over time, but we can also relate it to displacement using the chain rule of calculus. We know that:
\( a = v \cdot \frac{dv}{ds} \)
Substituting the values we have:
- Velocity \( v = 20 \, \text{m/s} \)
- Rate of change of velocity \( \frac{dv}{ds} = -8 \, \text{m/s/m} \)
Now, substituting these values into the equation for acceleration:
\( a = 20 \, \text{m/s} \cdot (-8 \, \text{m/s/m}) \)
Calculating this gives:
\( a = -160 \, \text{m/s}^2 \)
Interpreting the Result
The negative sign indicates that the acceleration is acting in the opposite direction to the velocity, which means the particle is decelerating. Therefore, we can conclude that the acceleration is:
- 160 m/s² decreasing with time
Final Answer
Based on the options provided, the correct choice is:
a. 160 m/s² decreasing with time
This analysis shows how the relationship between velocity, displacement, and acceleration can be utilized to find the acceleration of a particle in motion. Understanding these concepts is crucial in physics, especially when dealing with motion in a straight line.
