P01-090 – Relationship Between Velocity & Acceleration

Relationship Between Velocity and Acceleration

Effect of acceleration on velocity:

In this paragraph, we summarize the important results derived from defining instantaneous acceleration and instantaneous velocity. In particular, we focus on the effect of instantaneous acceleration on instantaneous velocity.

Consider an object moving along the $x$-axis in the positive direction following the different scenarios below.

Scenario 1: velocity $v_x$ and acceleration $a_x$ are both positive

If $v_x$ and $a_x$ are both positive, the particle speeds up in the positive direction.

Scenario 2: velocity $v_x$ and acceleration $a_x$ are both negative

If $v_x$ and $a_x$ are both negative, the particle speeds up in the negative direction.

Scenario 3: velocity $v_x$ is positive and acceleration $a_x$ is negative

If $v_x$ is positive and $a_x$ is negative, the particle slows down as it moves to the right.

Note: given enough time, the particle could come to a full stop before speeding up in the negative direction (assuming the acceleration continues to act long enough on the particle).

Scenario 4: velocity $v_x$ is negative and acceleration $a_x$ is positive

If $v_x$ is negative and $a_x$ is positive, the particle slows down as it moves to the left.

Note: given enough time, the particle could come to a full stop before speeding up in the positive direction (assuming the acceleration continues to act long enough on the particle).

Scenario 5: acceleration $a_x$ is zero

If the acceleration $a_x$ is zero, the particle moves at constant speed (uniform linear motion).

Conclusion:

The above scenarios illustrate the following conclusion

Acceleration is a quantity that changes the velocity vector, in magnitude or direction, over time.