How to Determine Initial Velocity Without Acceleration or Final Velocity

Understanding the relationships between initial velocity, final velocity, distance, time, and acceleration—collectively known as the SUVAT equations—is crucial in resolving various kinematics problems. However, what do you do when acceleration or final velocity are unknown? This article explores the methods to find the initial velocity of an object in motion without these key pieces of information.

The SUVAT Equations: A Brief Recap

The SUVAT equations are a set of formulas that describe uniformly accelerated motion. These equations relate initial velocity (U), final velocity (V), acceleration (A), time (T), and distance (S) as follows:

S UT ?AT2 V U AT V2 U2 2AS S (U V)T/2 S V T - ?AT2

When Acceleration or Final Velocity is Unknown

Typically, the SUVAT equations require at least two of the five parameters to determine the unknown. However, if acceleration (A) or the final velocity (V) is not known, we can use specific relationships to find the initial velocity (U).

1. Using Distance and Time

One method involves using the equation S (U V)T/2 when the distance (S) and time (T) are known. However, without the final velocity (V), we need an alternative approach. The equation S UT ?AT2 can be rearranged to find U without A if V is unknown:

S UT ?AT2

Rearranging for U when A is unknown:

U (S - ?AT2) / T

2. Estimation and Guessing

Another simplistic and often less reliable method involves making an educated guess about the initial velocity. While this approach lacks precision, it can offer a reasonable estimate based on real-world observations and known properties of the object in motion.

Real-World Applications and Examples

Imagine you are observing a ball in motion and want to determine its initial velocity without knowing its acceleration or final velocity. Consider the following scenarios:

Scenario 1: Ball Thrown Vertically

Suppose you throw a ball straight up and measure the maximum height it reaches (S) and the time (T) it takes to reach the top. Knowing that gravitational acceleration (A -g, where g ≈ 9.81 m/s2) is constant, you can use the displacement-time relationship:

S UT - ?gT2

Rearranging this equation to solve for U:

U (S ?gT2) / T

Scenario 2: Projectile Motion

In projectile motion, the horizontal velocity remains constant if we ignore air resistance. If you measure the horizontal distance (S) and the time (T) it takes for a projectile to travel this distance, the horizontal initial velocity (Ux) can be found using:

S UxT

Rearranging for Ux when A is zero:

Ux S / T

Conclusion

While the methods described above can help in finding the initial velocity without knowing the acceleration or final velocity, the accuracy depends on the assumptions and available data. In real-world scenarios, it is often prudent to apply the SUVAT equations where possible, making use of the additional parameters available.

Keywords: initial velocity, acceleration, final velocity, suvat equations