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Q3.11(d):
Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (d) with positive value of acceleration must be speeding up.

Solution :

Step 1: Theoretical Foundation

The state of a particle's motion is determined by the relationship between the velocity vector ($\vec{v}$) and the acceleration vector ($\vec{a}$). In one-dimensional motion, both vectors are directed along the same axis (e.g., the $x$-axis).

  • Speeding up occurs when the velocity and acceleration vectors point in the same direction ($\vec{v} \cdot \vec{a} > 0$).
  • Slowing down (deceleration) occurs when the velocity and acceleration vectors point in opposite directions ($\vec{v} \cdot \vec{a} < 0$).

Step 2: Analytical Evaluation of the Statement

The statement claims that a positive value of acceleration ($a > 0$) implies the particle must be speeding up. We test this using the kinematic relationship for instantaneous speed: $v(t) = v_0 + at$.

Consider two distinct scenarios where $a > 0$:

Scenario Velocity ($v$) Acceleration ($a$) Effect on Speed ($|v|$)
I $v > 0$ $a > 0$ Speeding up (Magnitude $|v|$ increases)
II $v < 0$ $a > 0$ Slowing down (Magnitude $|v|$ decreases)

Step 3: Justification and Counter-Example

Verdict: False.

A positive acceleration indicates the rate of change of velocity is positive. If the particle is moving in the negative direction (e.g., $v = -10 \, \text{m/s}$) and experiences a positive acceleration (e.g., $a = +2 \, \text{m/s}^2$), the velocity will change from $-10$ to $-8$, then to $-6$. [By the definition of vector addition in kinematics].

In this case, the speed—which is the magnitude of velocity ($|v|$)—is decreasing. Thus, positive acceleration does not guarantee an increase in speed.

x v = -10 m/s a = +2 m/s²

Step 4: Conclusion

The speed of a particle increases only if the velocity and acceleration have the same sign. If $a > 0$ and $v < 0$, the particle is decelerating (slowing down). Therefore, the statement is scientifically inaccurate.

Final Solution: The statement is False. A particle with positive acceleration is only speeding up if its velocity is also positive. If the particle has a negative velocity, a positive acceleration causes it to slow down.


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