Simulation

Uniformly Accelerated Motion

Video Lesson

Lesson objective

Dear learners,

At the end of the lesson you will be able to

• Explain a uniformly accelerated motion in 1D;
• Explain the difference between average velocity and instantaneous velocity; and
• Solve problems involving average velocity, instantaneous velocity and acceleration

Brainstorming Question

• How will it be possible for a car to move but not accelerating?

Key Terms and Concepts

LESSON Presentation

Acceleration

• Acceleration :- is the rate of change of velocity or change in velocity per unit time.
• An object is said to accelerate when its velocity changes in magnitude, in direction or both in magnitude and direction.
• Acceleration is a vector quantity, and
• The SI unit of acceleration is m/s. 2
• Negative acceleration may or may not be deceleration, and deceleration may or may not be considered negative acceleration.
• If the velocity and acceleration have the same sign whether positive or negative it is speeding up.
• If the velocity and acceleration have different signs, then the object is slowing down.
• Acceleration can be positive, negative or zero. 
• When an object’s velocity increases with time, it can be termed Positive Acceleration. 
• When an object’s velocity decreases with time, it can be termed Negative Acceleration or Retardation.
• When the velocity is zero, it is termed Zero Acceleration.
• A few examples of acceleration are the falling of an apple, the moon orbiting around the earth, or when a car is stopped at the traffic lights.
• Through these examples, we can understand that when there is a change in the direction of a moving object or an increase or decrease in speed, acceleration occurs.
• $accleration=\frac{\left( final velocity \right)-\left( initial velocity \right)}{time}$
• $accleration=\frac{Change in velociy}{time}$
• $\overrightarrow{a}=\frac{\Delta V}{t}$

Where,

• a = is the acceleration in m.s^-2
• v_f  =s the final velocity in m.s^-1
• v_i  = is the initial velocity in m.s^-1
• t = is the time interval in s
• Δv = is the small change in the velocity in m.s^-1

Figure 3.1 A car slowing down and speeding up

Average acceleration

The average acceleration :- the total change in velocity in the given interval divided by the total time taken for the change.

For a given interval of time, it is denoted as $\overrightarrow{a}$

Mathematically,

$\overrightarrow{a}=\frac{\Delta V}{t}=\frac{V_{f}-V_{i}}{t_{f}-t_{i}}$

Examples 3.1

A truck accelerates from 6 m/s to 10 m/s in a time period of 10 s. What will be its acceleration?

Solution

• Initial Velocity (u) = 6 m/s
• Final Velocity (v) = 10 m/s
• Time taken (t) = 10 s

Using the Acceleration Formula, 

Acceleration a = (v – u) / t

a = (10 m/s – 6 m/s) / 10 s

a = 0.4 m/s²

Thus, the acceleration of the truck is calculated as 0.4 m/s².

Instantaneous Acceleration

Instantaneous acceleration:-is the rate at which velocity changes using both speed and direction with respect to time such that the time interval goes to zero.

Example 3.2

The position of a particle is x(t) = 2t + 0.7t³ m. Find the instantaneous acceleration at t= 3 sec?

Solution

x(t) = 2t + 0.7t³

So,

v(t) = dx(t)/dt = 2+2.1t² m/s

Now, 

a(t) = dv(t)/dt = 4.2t m/s²

Therefore,

At t = 3 sec instantaneous acceleration is;

4.2t = 4.2 × 3= 12.6 m/s² (Answer)

Examples 3.3

A particle is in motion and is accelerating. The position of the velocity is v(t) = 10t – 3t² m/s

• Find the functional form of acceleration.
• Find the instantaneous velocity at t= 1,3,4,5 s
• Find the instantaneous acceleration at t= 1,3,4,5s.
• Analyze the results of © in terms of acceleration and velocity vectors directions.

Solution

   v(t) = 10t – 3t² m/s

1. a(t) = dv(t)/dt = 10 – 6t m/s²
2. v(1s) = 7 m/s

v(3s) = 3 m/s

v(4s) = -8 m/s

v(5s) = -25 m/s

a(1s) = 4 m/s²

Motion with Constant Acceleration

• If an equal amount of velocity increases in equal intervals of time, then the object is said to be in uniform acceleration motion
• In this case, the average acceleration equals the instantaneous acceleration, and the average velocity is the average of the initial and final velocities.

$V_{av}=\frac{V_{i}+V_{f}}{2}$

• If the velocity of the object changes by unequal amounts in equal intervals of time, the object is said to be non-uniform acceleration.