To find the acceleration of the particle at the given instant, we can use the relationship between velocity, displacement, and acceleration. The problem states that the velocity of the particle is decreasing at a rate of 8 m/s for every meter of displacement. This means that the acceleration can be expressed in terms of the velocity and the displacement.
Understanding the Relationship
We know that acceleration (a) can be defined as the rate of change of velocity (v) with respect to time (t). However, in this case, we have a situation where the velocity is changing with respect to displacement (s). The relationship can be expressed as:
- Acceleration (a) = -8 m/s² (since the velocity is decreasing)
- Velocity (v) = 20 m/s
- Displacement (s) = 14 m
Using the Given Rate of Change
The problem states that the velocity decreases at a rate of 8 m/s for every meter of displacement. This implies that:
a = -8 * (dv/ds)
Here, dv/ds is the rate of change of velocity with respect to displacement. Since the velocity is decreasing, we can express this as:
dv/ds = -8 m/s per meter
Calculating the Acceleration
Now, we can substitute this value into our equation for acceleration:
a = -8 * (-8)
Calculating this gives:
a = 64 m/s²
Final Thoughts
Thus, the acceleration of the particle at the instant when the velocity is 20 m/s and the displacement is 14 m is 64 m/s². This positive value indicates that the particle is experiencing a deceleration, as it is slowing down, but the magnitude of the acceleration is significant.
