Problem 18 - Basic Engineering Mathematics

Question

Velocity \(v\) is given by \(v=u+a t\). If \(u=9.86 \mathrm{~m} / \mathrm{s}, a=4.25 \mathrm{~m} / \mathrm{s}^{2}\) and \(t=6.84 \mathrm{~s}\), find \(v\), correct to 3 significant figures.

Step-by-Step Solution

Answer

The velocity \( v \) is \( 38.9 \, \text{m/s} \).

Step 1: Understand the Equation

The equation given is the formula for velocity in uniform acceleration motion: \( v = u + a t \). This shows the final velocity \( v \) as a combination of the initial velocity \( u \), the acceleration \( a \), and the time \( t \).

Step 2: Identify Known Values

We're provided with the initial velocity \( u = 9.86 \, \text{m/s} \), acceleration \( a = 4.25 \, \text{m/s}^2 \), and time \( t = 6.84 \, \text{s} \). These values will be substituted into the equation to find \( v \).

Step 3: Substitute and Calculate

Substitute the known values into the equation: \( v = 9.86 + 4.25 \times 6.84 \). First, calculate \( 4.25 \times 6.84 \) which equals \( 29.07 \). Then add this to \( 9.86 \) to find \( v = 38.93 \).

Step 4: Round to Significant Figures

The problem requires rounding the result to 3 significant figures. \( 38.93 \) rounded to 3 significant figures is \( 38.9 \).

Key Concepts

Uniform Acceleration

Uniform acceleration refers to a constant rate of change of velocity over time. This concept is central in physics, especially when analyzing motion. In situations where acceleration is uniform, the velocity of an object changes by the same amount each second.
This makes the motion predictable, allowing us to use equations like the one in the exercise:

• \( v = u + a t \), where

\( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the constant acceleration, and \( t \) is the time over which the acceleration occurs.
Understanding uniform acceleration helps in solving motion problems efficiently, as you can calculate any of the variables if the others are known.

Initial Velocity

Initial velocity, often symbolized as \( u \) in equations, is the velocity of an object at the start of an observation. This value is crucial because it forms the baseline from which changes in velocity are measured.

• In our exercise, the initial velocity is given as \( u = 9.86 \, \text{m/s} \).
• It is used in the velocity equation to determine how much the object speeds up over time.

Knowing the initial velocity allows us to predict future velocities when we have uniform acceleration, as it significantly influences the final velocity calculation.

Acceleration

Acceleration is the rate at which an object changes its velocity. It is often represented by the letter \( a \) and expressed in \( \text{m/s}^2 \). Acceleration can be positive, meaning an increase in speed, or negative, indicating deceleration.
In uniform motion problems, the constant acceleration simplifies calculations, as seen in the example equation \( v = u + a t \). Here, the acceleration \( a \) is a constant value needed to calculate the change in velocity over a specific time.

• In our worked example, \( a = 4.25 \, \text{m/s}^2 \).

This constant is crucial for calculating how much faster (or slower) an object moves after a given time has passed.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. These figures include all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal part.

• When performing calculations, it's vital to follow significant figure rules to ensure precision is maintained and conveyed accurately.
• In the exercise, the final result was rounded to three significant figures to match the precision required.

By rounding \( 38.93 \) to \( 38.9 \), the precision aligns with the least precise measurements given initially, which is essential in scientific and technical fields to communicate measurement uncertainty clearly.