Non-uniform accelerated motion

Non-uniform Accelerated Motion

In physics, motion refers to the change in position of an object over time. When an object is in motion, it can have different types of motion, such as uniform motion or non-uniform motion. In this article, we will focus on non-uniform accelerated motion.

Definition

Non-uniform accelerated motion refers to the motion of an object where its velocity changes at a non-constant rate. This means that the object's speed and/or direction of motion is changing over time. The rate of change of velocity is called acceleration.

Acceleration

Acceleration is defined as the rate of change of velocity with respect to time. It is a vector quantity, meaning it has both magnitude and direction. The formula for acceleration is:

$$a = \frac{{\Delta v}}{{\Delta t}}$$

where:

  • $a$ is the acceleration
  • $\Delta v$ is the change in velocity
  • $\Delta t$ is the change in time

Acceleration can be positive or negative, depending on whether the object is speeding up or slowing down. If the acceleration is positive, the object is said to be undergoing positive acceleration or speeding up. If the acceleration is negative, the object is said to be undergoing negative acceleration or slowing down.

Differences between Uniform and Non-uniform Accelerated Motion

Uniform Accelerated Motion Non-uniform Accelerated Motion
Velocity changes at a constant rate Velocity changes at a non-constant rate
Acceleration is constant Acceleration is not constant
Object covers equal distances in equal intervals of time Object covers unequal distances in equal intervals of time
Object covers equal distances in unequal intervals of time Object covers unequal distances in unequal intervals of time
Examples: Free fall of an object, motion of a car with constant acceleration Examples: Throwing a ball upwards, motion of a car with varying acceleration

Equations of Motion

In non-uniform accelerated motion, we can use the equations of motion to describe the motion of an object. These equations relate the displacement, initial velocity, final velocity, acceleration, and time.

  1. First Equation of Motion:

$$v = u + at$$

where:

  • $v$ is the final velocity
  • $u$ is the initial velocity
  • $a$ is the acceleration
  • $t$ is the time

This equation relates the final velocity of an object to its initial velocity, acceleration, and time.

  1. Second Equation of Motion:

$$s = ut + \frac{1}{2}at^2$$

where:

  • $s$ is the displacement
  • $u$ is the initial velocity
  • $a$ is the acceleration
  • $t$ is the time

This equation relates the displacement of an object to its initial velocity, acceleration, and time.

  1. Third Equation of Motion:

$$v^2 = u^2 + 2as$$

where:

  • $v$ is the final velocity
  • $u$ is the initial velocity
  • $a$ is the acceleration
  • $s$ is the displacement

This equation relates the final velocity of an object to its initial velocity, acceleration, and displacement.

Example

Let's consider an example to understand non-uniform accelerated motion better.

Example: A car starts from rest and accelerates uniformly at a rate of 2 m/s^2 for 10 seconds. After that, it decelerates uniformly at a rate of 1 m/s^2 for 5 seconds. Calculate the final velocity and displacement of the car.

Solution:

Given:

  • Initial velocity ($u$) = 0 m/s
  • Acceleration ($a$) = 2 m/s^2 (for 10 seconds)
  • Deceleration ($a$) = -1 m/s^2 (for 5 seconds)
  1. Calculate the final velocity during the acceleration phase using the first equation of motion:

$$v = u + at$$ $$v = 0 + (2 \times 10)$$ $$v = 20 \, \text{m/s}$$

  1. Calculate the displacement during the acceleration phase using the second equation of motion:

$$s = ut + \frac{1}{2}at^2$$ $$s = 0 + \frac{1}{2}(2 \times 10^2)$$ $$s = 100 \, \text{m}$$

  1. Calculate the final velocity during the deceleration phase using the first equation of motion:

$$v = u + at$$ $$v = 20 + (-1 \times 5)$$ $$v = 15 \, \text{m/s}$$

  1. Calculate the displacement during the deceleration phase using the second equation of motion:

$$s = ut + \frac{1}{2}at^2$$ $$s = 20 \times 5 + \frac{1}{2}(-1 \times 5^2)$$ $$s = 75 \, \text{m}$$

  1. Calculate the total displacement of the car:

$$\text{Total displacement} = \text{Displacement during acceleration} + \text{Displacement during deceleration}$$ $$\text{Total displacement} = 100 + 75$$ $$\text{Total displacement} = 175 \, \text{m}$$

  1. Calculate the final velocity of the car:

$$\text{Final velocity} = \text{Final velocity during deceleration}$$ $$\text{Final velocity} = 15 \, \text{m/s}$$

Therefore, the final velocity of the car is 15 m/s and the total displacement is 175 m.

Conclusion

Non-uniform accelerated motion refers to the motion of an object where its velocity changes at a non-constant rate. The rate of change of velocity is called acceleration. In non-uniform accelerated motion, the object's speed and/or direction of motion is changing over time. We can use the equations of motion to describe the motion of an object in non-uniform accelerated motion. These equations relate the displacement, initial velocity, final velocity, acceleration, and time. By using these equations, we can calculate the final velocity and displacement of an object in non-uniform accelerated motion.