Understanding Time and Initial Velocity in Motion Analysis

When dealing with the motion of objects, it is often tempting to assume that the initial time t0 in a problem is set to zero. While this can be a convenient convention, it is not always practical or necessary.

Can You Assume t is Always 0 for Initial Velocity?

It is important to understand that you cannot simply assume that the initial time t for the initial velocity of an object in motion is always zero. This is a choice that you can make, but it is not always the most suitable or necessary for solving a given problem.

For instance, when a car is driving down a road, it encounters a red light and stops. When the light turns green, the car accelerates and travels at a speed limit. This scenario involves multiple stages, each with its own initial time. In such cases, setting t to zero at the start of each stage may not be feasible.

Time Translation Symmetry and Time Invariance

However, if you consider a new time variable t', defined as t' t - t0, you can always reset the time reference to zero at the start of your chosen stage. This does not change the underlying physical laws, which remain invariant under time translation symmetry. This means that the rules governing the motion of an object do not change with time, a property related to the conservation of energy.

Setting Initial Conditions in Physics Problems

The initial conditions in a physics problem, including the initial time and velocity, must be specified based on the specific problem you are trying to solve. For example, when setting up the problem, you can choose to let t 0 at any point in time you desire. However, the initial velocity, being a part of the problem itself, must be set according to the specific conditions given in the problem statement.

Mathematical Implications of t 0

Mathematically, if t 0, then the equation for displacement (s) in relation to velocity (v) and time (t) is given by:

s vt

Substituting t 0 into the equation for displacement gives:

s v(0)

s 0

Thus, if the initial time is set to zero, the displacement is also zero. This is a direct consequence of the chosen initial time and does not reflect the physical state of the object in motion.

Conclusion

In summary, while you can choose the initial time t to be zero for convenience, it is not a universal rule. The choice depends on the specific problem and its requirements. Understanding the principles of time translation symmetry and time invariance can help you better manage and solve motion problems.