A Body is Accelerated Continuously What is the Form of the Graph
Acceleration is defined as the rate of change of velocity of an object concerning time. The object's orientation depends upon the acceleration of the object as well as the speed acting on it. A simple example in our daily life is when a car speeds up, when the car slows down, when the car turns here the speed of the car changes which is indirectly called acceleration.
Uniform Acceleration is nothing but this rate of change of speed is defined as Uniform but how can it be uniform? Is that possible? Let us go through this one to clear all our doubts on uniform Acceleration.
What is Uniform Acceleration?
If any object's speed or velocity is increasing at a constant rate then we can say that the body has a constant acceleration or change of equal velocity in equal Intervals of time.
a= dv/dt
a = acceleration of body
dv= rate of change of velocity
dt= rate of change of time
SI Unit = meter/square second
Ex : Motion of a free falling body :
In the case of a free falling body, its speed gradually decreases in equal intervals of time and finally becomes zero hence it is an example of uniform acceleration.
Motion in a Straight Line: Important Questions
Equations of Uniformly Accelerated Motion
There are 3 main equations for the calculation of uniformly accelerated motion
- Velocity equation
- Distance equation
- Velocity and distance relation equation
Velocity Equation
Consider the graph in which object D has an initial velocity u after a certain time it moved through a distance now in time t
Now from the above graph let us consider
OD=u
OE=t
OV =v
BE = BA+AE
Now let us consider OV= CD+OD
From above we can write as v= CD + uà (1)
Form equation of acceleration a=(v-u)/t
a = OV-OD/t
from 1
a = v-u/t
at = v-u
v = u+ at
where v= final velocity
u= initial velocity
a = acceleration
t = time
Hence the Velocity equation for uniform acceleration is derived.
Distance Equation
Distance equation is very important for the determination of uniform Acceleration
From the above graph let us consider an trapezium DBEO
Distance travelled by object = Area of Trapezium DBEO
=Area of rectangle DAEO+ Area of Triangle DBA
=OD*OE+ (1/2)*DA*AB
= ut+1/2 *(v-u)
From the velocity equation, the above can be written as
S= ut+1/2*t*at
S =ut+(1/2)*a*t2
Where s= distance traveled by an object
u= initial velocity
a= acceleration
t = time required for travel
Hence the distance equation for uniform acceleration is determined.
Velocity and Distance Equation Relation
For now we have derived equations for velocity and distance so let us now derive the relation between them.
Let us use the same graph to draw a conclusion
In this case also we have to find out the distance travelled by the body using the above graph
Distance travelled by the bodies = area of trapezium DBEO
= 1/2 * (sum of parallel sides)*(height)
= ½ *(OD+BE)*OE
= (u +v)/2*t
= 1/2(u + v)*t
= 1/2(v + u)(v-u)/a
2as = v2-u2
V 2 = u 2 +2as
Where as v= final velocity
U= initial velocity
S= distance
a = acceleration
Hence we derived the relation between velocity and distance using graph equations.
Things to Remember based on Uniformly Accelerated Motion
- Acceleration refers to the rate of change of velocity with respect to time.
- Uniform Acceleration refers to a phenomenon where the acceleration of an object remains constant throughout a given time.
- Examples of Uniform Acceleration include a ball being dropped from a high location, brakes being applied to a bicycle, a ball rolling down a slope, etc.
- Uniformly Accelerated Motion can be defined through three kinematic equations: Velocity Equation, Distance Equation, Distance and Velocity Equation Relation.
Sample Questions based on Uniformly Accelerated Motion
Ques. A car from rest travels a distance of 40m at a constant acceleration of 20m/s. How long does it take to cover the distance? (4 marks)
Ans. From the above problem,
Given s=40m
a= 20m/s
t= ?
u=0
s= distance travelled by the body
u=initial velocity
a= acceleration of the car
t = time
s=u t+(1/2)a t2
= (0)t+0.5 *20*t2
40 = 0.5*20* t2
40/(10) = t2
t2 = 4
t= 2
Therefore 2 seconds are required to cover a distance of 40m with an acceleration of 20m/s2.
Ques. A man starts at rest, travels a distance of 40min 4 secs to determine the acceleration at which he travels? (4 marks)
Ans. From the above
Given s =40m
t= 4 sec
u=0
a= ?
s = ut+1/2at2
40 =(0)t +0.5*a*16
40 = 8a
a = 40/8
a= 5m/s2
Therefore the man travels with a constant acceleration of 5m/s2
Ques. A body moves from rest and travels with a constant acceleration of 2m/s 2 Find its speed and distance after 5 secs. (3 marks)
Ans. u=0,t=5,a=2
V= u+ at
V= 0+2*5
V= 10m/s
S= ut+0.5 a t2
S = (0)5 + 0.5*2*25
S= 25m
So the body traveled with a speed of 10m/s to a distance of 25m.
Ques. A car begins at a speed of 3m/s and accelerates at a rate of 2m/s 2 over a distance of 50m. Find the final speed of the car? (3 marks)
Ans. From above
Given u=3m/s
a= 2m/s2
s=50m
v= ?
v2-u2=2as
v2-9 = 2*2*50
v2 = 200+9
v= 14.49m/s
So the final velocity of the car which begins at a speed of 3m/s and accelerates at a rate of 2m/s2 is 14.49m/s.
Ques. Calculate the initial velocity of a car with a final speed of 50m/s accelerates at an acceleration of 2m/s 2 ? (3 marks)
Ans. From the above problem,
Given a= 2m/s2
V= 50m/s
V= u + at
50 = u+2*50
50 = u+100
U = 50/100
U =0.5m/s
The initial velocity of the car is 0.5m/s
Ques. Give 4 examples of uniform Acceleration you see in daily life? (4 marks)
Ans. Examples of uniform acceleration are:
- Ball rolling down from a slope
- Sky driver jumping out of a plane
- Bicycle when brakes are jammed
- Ball dropped from a building
Ques. Is retardation a uniform acceleration? (2 marks)
Ans. Retardation is known as the inverse of the acceleration denoted as
R= (-a).
Ques. What is the difference between uniform acceleration and Non-uniform acceleration? (1 mark)
Ans. In uniform acceleration velocity of the body changes in equal intervals of time whereas in non-uniform acceleration it doesn't occur in equal intervals of time.
Source: https://collegedunia.com/exams/uniformly-accelerated-motion-constant-acceleration-equation-derivation-physics-articleid-968
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