An Introduction to Uniformly Accelerated Motion
We all know that acceleration is the rate at which the velocity of an object changes with respect to time. However, do you know what is uniformly accelerated motion? It can be difficult to perceive how an aspect that is defined by a rate of change can be defined as uniform. Therefore, let us study such uniformity in acceleration in detail, especially since this a vital concept that is necessary for numerous other chapters too.
Define Uniform Acceleration
One can say that uniform acceleration definition refers to an acceleration of an object, which remains constant irrespective of time. In simpler terms, a number equal to the acceleration in such a motion does not change as a function of time.
Some uniformly accelerated motion examples include a ball rolling down a slope, a skydiver jumping out of a plane, a ball dropped from the top of a ladder and a bicycle whose brakes have been engaged.
Keep in mind that these examples of uniform application do not maintain absolute uniformity of acceleration, due to the interference of gravity and/or friction. However, these are still some of the cases where acceleration would be uniform if gravitational force and friction is considered zero.
Equations of Uniformly Accelerated Motion
After understanding what is uniform acceleration, one should proceed to learn about the three kinematic equations that define such a motion.
1. Velocity Equation
To understand the first equation, refer to the graph below.|
(Image will be Uploaded Soon)
Let us assume that the initial velocity of an object is u. Similarly, uniform acceleration for this object is a. Now, the object reaches at point B after time t, post which, its velocity becomes v.
DA represents a parallel line to the x-axis from the point where the object started moving. BA is a line parallel to the y-axis which is connecting A to the end-point of the body. This vertical line meets the x-axis at point E. Consider OD as u.
Suppose OE = t or time
From this graph, we can derive
BE = AB + AE
v = DC + OD
From the graph, it is apparent that DC = AB and OD = AE (OD is also u)
v = DC + u (i)
Now, we know acceleration (a) = \[\frac {(v-u)} {t}\]
Thus, a = \[\frac {OC-OD} {t}\]
a = \[\frac {DC} {t}\]
at = DC (ii)
Substituting the value of DC from (ii) to (i)
V = at + u
This is the first kinematic equation for uniformly accelerated motion.
2. Distance Equation
The key to define uniform acceleration is the second equation to determine distance.
Distance (s) = Area of ABD + Area of ADOE
s = \[\frac {1} {2}\] x AB x AD + (OD x OE)
s = \[\frac {1} {2}\] x DC x AD + (u + t)
AB = DC
s = \[\frac {1} {2}\] x at x t + ut
s = \[\frac {1} {2}\] at2 + ut
or, s = ut + \[\frac {1} {2}\] at2
3. Equation to Relate Distance and Velocity
The third and final equation to define uniform acceleration is the one that relates distance (s) and velocity (v).
Area of trapezium ABDOE = ½ x (sum of parallel sides – distance between parallel sides)
s = \[\frac {1} {2}\] (DO + BE) x OE
Therefore, s = \[\frac {1} {2}\](v + u) x t …….(iii)
From equation (ii), we know a = \[\frac {(v-u)} {t}\]
t = \[\frac {(v-u)} {a}\] …….(iv)
Therefore, s = \[\frac {1} {2}\] (v + u) x (v – u)/a
s = \[\frac {1} {2}\]a (v + u)(v – u)
2as = v2 – u2
v2 = u2 + 2as
Thus, this is how we write the three equations of uniformly accelerated motion.
FAQs on Uniformly Accelerated Motion
1. What is Uniformly Accelerated Motion (UAM), and what does 'uniform acceleration' mean?
Uniformly Accelerated Motion, often abbreviated as UAM, describes the motion of an object where its velocity changes by an equal amount in every equal interval of time. The core of this concept is 'uniform acceleration,' which means the acceleration of the object is constant; it does not change with time. An object in UAM will either speed up or slow down at a steady rate.
2. What are the three main equations of uniformly accelerated motion?
The three kinematic equations that describe uniformly accelerated motion are fundamental for solving problems in physics. They relate initial velocity (u), final velocity (v), acceleration (a), time (t), and displacement (s). The equations are:
- First Equation (Velocity-Time Relation): v = u + at
- Second Equation (Position-Time Relation): s = ut + ½at²
- Third Equation (Position-Velocity Relation): v² = u² + 2as
3. What are some real-world examples of uniformly accelerated motion?
Uniformly accelerated motion is an idealisation, but many real-world scenarios approximate it closely. Key examples include:
- An object in free fall near the Earth's surface (e.g., a dropped apple), where it accelerates downwards at a constant rate 'g' (approx. 9.8 m/s²), assuming air resistance is negligible.
- A ball rolling down a smooth, inclined plane.
- A vehicle moving in a straight line while applying a constant braking force, causing uniform deceleration.
4. How are position, velocity, and acceleration represented graphically for an object in UAM?
For an object in uniformly accelerated motion, the graphs have distinct shapes:
- Position-Time Graph: A parabola (a curve). The slope of the curve at any point gives the instantaneous velocity.
- Velocity-Time Graph: A straight line with a non-zero slope. The slope of this line represents the constant acceleration, and the area under the graph gives the displacement.
- Acceleration-Time Graph: A horizontal straight line, indicating that the acceleration is constant and does not change over time.
5. Is uniformly accelerated motion the same as uniform motion?
No, they are fundamentally different concepts. Uniform motion means an object moves with constant velocity (and therefore, zero acceleration). In contrast, uniformly accelerated motion means the object's velocity is constantly changing at a steady rate (i.e., it has a non-zero, constant acceleration). The 'uniform' part in UAM refers to the acceleration, not the velocity.
6. What is the key difference between uniformly and non-uniformly accelerated motion?
The key difference lies in the nature of the acceleration. In uniformly accelerated motion, the acceleration is constant. In non-uniformly accelerated motion, the acceleration itself changes with time. For example, a car moving in city traffic constantly speeding up and slowing down by different amounts is an example of non-uniform acceleration, and the standard kinematic equations (like v = u + at) do not apply directly.
7. Under what conditions can we apply the equations of uniformly accelerated motion?
The three main equations of motion are valid only when one crucial condition is met: the acceleration of the object must be constant throughout its motion. This applies to both magnitude and direction. In many textbook problems, we also make simplifying assumptions, such as neglecting air resistance, to ensure the acceleration (like 'g' in free fall) remains constant.
8. Can an object have zero velocity but still be in uniformly accelerated motion? Explain with an example.
Yes, an object can have an instantaneous velocity of zero while still accelerating. The classic example is a ball thrown vertically upwards. At the very peak of its trajectory, the ball momentarily stops and has a velocity of zero. However, the force of gravity is still acting on it, causing a constant downward acceleration of approximately 9.8 m/s². Therefore, it is still in uniformly accelerated motion.
9. How do the directions of velocity and acceleration affect uniformly accelerated motion?
The relative directions of velocity and acceleration determine whether an object speeds up or slows down:
- If the velocity and acceleration vectors point in the same direction, the object's speed increases.
- If the velocity and acceleration vectors point in opposite directions, the object's speed decreases. This is often referred to as deceleration or retardation.
