Problem 9 - Applied Mathematics

Question

(a) A particle is moving in a straight line with constant acceleration. (b) The average velocity of a particle moving in a straight line is the algebraic average of the initial and final velocities.

Step-by-Step Solution

Answer

The average velocity is the mean of the initial and final velocities: \[ v_{avg} = \frac{u + v}{2} \].

Step 1: Understanding Constant Acceleration

Constant acceleration means the velocity of the particle changes at a constant rate. The basic kinematic equation to describe this motion is: \[v = u + at\] where:\( v \) = final velocity,\( u \) = initial velocity,\( a \) = acceleration,\( t \) = time.

Step 2: Average Velocity Calculation

The average velocity \( v_{avg} \) for a particle moving in a straight line with constant acceleration is the mean of the initial and final velocities. This can be given by the formula: \[ v_{avg} = \frac{u + v}{2} \]Where: \( u \)= initial velocity, \( v \)= final velocity.

Step 3: Combining the Equations

If the initial velocity \( u \) and the final velocity \( v \) are required to find the average velocity, use the equation for constant acceleration: \[ v = u + at \]Once we have the final velocity, insert it into the average velocity formula above.

Key Concepts

kinematic equations

Kinematic equations are used to describe the motion of objects. They are especially useful when dealing with situations of constant acceleration. These equations relate initial velocity, final velocity, acceleration, time, and displacement. Here are the basic kinematic equations you should know:

• 1. Equation for velocity: \[ v = u + at \] where 1. 2. Equation for displacement: \[ s = ut + \frac{1}{2}at^2 \] where
  
  • s = displacement,
  • ut = initial velocity multiplied by time,
  • \frac{1}{2} a t^2 = half the acceleration multiplied by the square of time.
  It’s essential to note that the equations assume acceleration is constant throughout the motion. Use them effectively for simplicity in solving complex motion problems.

average velocity

Average velocity is the mean velocity of an object in motion over a given time period. When dealing with constant acceleration, it is straightforward to calculate using the initial and final velocities.

• Formula: \[ v_{avg} = \frac{u + v}{2} \] where 1. 2. Formula: \[ v_{avg} = \frac{total \text{ displacement (x)}}{\text{total time (t)}} \] It gives us a simplification for calculating average velocity in uniformly accelerated motion. This formula is particularly useful since it blends both initial and final velocities into a single value when we need a representative measure of velocity over a time.

straight-line motion

Motion in a straight line refers to the movement of an object along a single axis. When analyzing such motion, it's significantly simpler than dealing with motion in a plane or space, since direction is constant. Characteristics of straight-line motion:

• 
  
  1.
  • 1. The velocity either increases or decreases linearly if there's constant acceleration.
    
    • 2.
      2. No change in direction simplifies calculations and understanding significantly.
      Comparing it to more complicated forms, straight-line motion is a fundamental building block to understanding and analyzing larger physics problems.