Kinematic Equations: The Ultimate Guide.
Kinematic Equations
Kinematic Equations. Kinematic is a branch of classical mechanics which is also known as Geometry of Motion. In this, we study the motion of two or more objects without a study that causes the object to move.
From this, we can Drive kinematic definition
“Kinematic is the Study of two or more object, point without considering the causes of these objects or points.”
The kinematic motion also studies objects or points differential angles, their mass, velocity, and acceleration. The kinematics is used to explain the movement of celestial objects and structures in astrophysics and to define the movement of joined-together device structures together electronic, robotic, and biomechanical engineering.
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Standardized physical research starts with kinematic studies. It is the Greek word “kinesis,” which means motion. It matches other English words such as kinesiology in which we study of human body motion and cinema. We can drive mathematical equations for kinematic motion by using the different aspects of movement, such as angle, time, displacement, acceleration.
Use of Kinematic Motion
We see the motion of objects in our daily life, such as football and car movement, from home to office. While we lay down, our heart moves blood through our veins. The non-living things have kinematic motion, such as the motion of their atoms or molecules. Water in the river, two objects collide with each other, is also an example of kinematic motion. Now the questions arise in our minds are: when the bat hits the ball at a certain angle, where will it hit the ground. How much time does the car take to move from home to office? If we understand this motion, it will help us to understand the other concept in Physics, such as acceleration.
When to use Kinematic Equations
We select the cinematic formula that involves both the unidentified variable that we are looking for and three of the kinematic variables which we already know. For the unknown we want to find, which will be the only unknown in the formula, we can solve that way.
We knew a ball on the ground was kicked forward with an initial velocity of V0=5 m/s, which it took a time interval t=3s for the ball to cover a displacement of Δx=8m We could use the kinematic formula Δx=v0t+at2, to solve for the unknown acceleration an of the ball algebraically. Consider the acceleration a was constant since we know every other variable in the formula besides a-Δx,v0t.
Solving tip: Notice that one of the five kinematic variables fails each kinematic formula
- v=v_{0}+at (Δx is missing in this equation)
- Δx =()t ( a is missing in this equation)
- Δx = v_{0}t+at^{2 } ( v is missing in this equation)
- v^{2}=+2aΔx ( t is missing in this equation)
To select the correct kinematic formula for your question, work out which variable you are not offered and don’t ask to find. For reference, in the above case, the ball’s final velocity (v) was neither given nor demanded, so we can choose a formula that does not involve velocity (v)
at all.
The kinematic formula Δx = V_{0}t+at^{2} is missing v, And it’s the best choice to solve the acceleration in this situation.
What are the Kinematic Equations?
The kinematic formula equations are often written as the following:
- v=v_{0}+at
- Δx =()t
- Δx = v_{0}t+at^{2}
- v^{2}=+2aΔx
Since the kinematic equations are valid even if the acceleration remains constant over the considered time, we must be not to use them when the acceleration varies. The kinematic formulae often suggest that all variables relate to the same direction: horizontal (x-axis), vertical (y-axis), etc.
It may seem that the assumption that the kinematic formulas function only for constant acceleration periods will seriously restrict the applicability of certain formulas.
How to solve Kinematic Equations?
There are four kinematic equations which associate with Displacement, Velocity, Time, and Acceleration.
- D= V_{t }+ 1/2 (at^{2})
- a=(v_{f}-v_{i})/t
- (v_{f}^{2}-v_{i}^{2})/2=D/t
- V_{f}^{2}=V_{f}^{2}+2aD
D= Displacement
a= Acceleration
t= time
V_{f}=Final velocity
V_{i}= Initial Velocity
Kinematic equation formula Questions.
Question# 1
Yousaf is riding his bicycle to the store at a velocity of 4 m/s when a car runs out in front of him. He quickly brakes to a complete stop, with an acceleration of – 2m/s2. What is his displacement?
Answer:
because Yousaf is stopped, the final velocity, vf = 0. His initial velocity, vi = 4 m/s. The acceleration, a = -2m/s2. Time is not given, so use equation (d) for displacement, D, because it is not time-dependent.
v_{f}^{2} = V_{i}^{2} + 2aD
(0)2= (4 m/s)2 +2(- 2 m/s2)D
0 = 16 m2/s2 + (- 4m/s2)D
-16 m2/s2 = (- 4 m/s2)D
16 m2/s2 = 4 m/s2)D
(16 m2/s2) / (4 m/s2) = D
Displacement= 4m
Question# 2
You travel at a constant velocity of 11 m/s for 5 minutes. How far have you travelled ?
Answer:
At constant velocity, vi = vf = 11 m/s. The time, t = 5 min, or t = (60 sec/min x 5 min) = 300 sec. Now use equation (b) to solve for displacement, D.
(v_{i} + v_{f})/2 = D/t
D = [(v_{i} + v_{f})/2] t
D = [(11 m/s + 11 m/s)/2] x 300 sec
D = (22 m/s)/2 x 300 sec
D = 11 m/s x 300 sec
D = 3,300 m The total displacement is 3, 300 m.
Question# 3
What is the acceleration of a car that speeds up from 11 m/s to 40 m/s after 10 seconds?
Answer:
The Vi = 11 m/s. The vf = 40 m/s. Time, t = 10 s. Use kinematic equation c) to solve for acceleration.
a = (v_{f} – v_{i})/t
a = (40 m/s – 11 m/s) /10 s
a = (29m/s)/10 s = 2.9 m/s^{2}
Question# 4
If a car accelerates at 3.0 m/s2 from a complete stop, how long will it take to go 3000 m?
Answer:
The acceleration, a = 2.9 m/s2, and the displacement, D = 3000 m. The car was at rest, so vi = 0. Use equation a) to solve for time.
D = v_{i}t + 1/2 at^{2}
3000 m = 0t + 1/2 (3.0 m/s^{2})t^{2}
3000 m = 1/2 (3.0 m/s^{2})/t^{2}
3000 m/ 1.5 m/s^{2} = t^{2}
2000 s^{2} = t^{2}
t = 44.72 sec
How to Drive Kinematic Equations?
Derivation of the first kinematic formula:
1. v=v_{0}+at ?
a = ( )
a = ( )
Finally, if we solve for v we get
V = v0+aΔt
And if we agree to use t for Δt, this becomes the first kinematic formula.
V = v0+at
2.Δx = ()t ?
Δx =t (v-)
If we distribute the factor of, we get:
Δx =vt –
We can simplify by combining the v_{0}, terms to get
Δx =vt –
We can simplify by combining the V0, terms to get
Δx = ()t
- Δx = v_{0}t+at^{2 }?
If we start with the second kinematic formula
= ()
and we use v=v_{0}+at to plugin for v, we get
=
++
And finally multiplying both sides by the time ttt gives us the third kinematic formula.
Δx = v_{0}t+at^{2 }
Again, we used other kinematic formulas, which have a requirement of constant acceleration, so this third kinematic formula is also only true under the assumption that the acceleration is constant.
- v^{2}=+2aΔx?
Δx=()t
We want to eliminate the time ttt from this formula. To do this, we’ll solve the first kinematic formula, v=v_{0}+at, for time to gett=If we plug this expression for time ttt into the second kinematic formula we’ll get
Δx=()()
Multiplying the fractions on the right-hand side gives
Δx= ()
And now solving for v^{2} we get the fourth kinematic formula.
Δx = v^{2}= +2aΔx
Velocity Equation
v^{2}= +2aΔx
Kinematic Equation of Circular Motion
The movement of objects moving in circles at a constant or uniform rate can be described by
- an instantaneous velocity, 𝑣, that is tangential to the circle or
- an angular velocity or angular frequency, 𝜔, that describes the rate of change of the angle with time.
The key point for an object going in a circle is that its speed remains the same but its velocity is always changing. Therefore the object is accelerating!
The speed around the circle and the angular frequency are related by:
𝑣=𝑟𝜔 or 𝜔=𝑣𝑟
The units of 𝜔 are rads−1, radians being dimensionless
The above figure shows a circle of radius 𝑟r with an arc of length 𝑙l subtending angle 𝜃θ at the center.
Arc length, radius, and angle are connected by 𝑙=𝑟𝜃l=rθ. [The angle in radians is 𝜃=𝑙𝑟θ=lr].
A differential length is connected to a differential angle similarly: d𝑙=𝑟d𝜃dl=rdθ.
The angle increases at rate 𝜔=d𝜃d𝑡ω=dθdt, with 𝜔ω the angular frequency or speed in radians per second.
The above gives us 𝑣=d𝑙d𝑡=𝑟d𝜃d𝑡=𝑟𝜔v=dldt=rdθdt=rω. Thus the speed around the circle and the angular frequency are related by:
𝑣=𝑟𝜔 or 𝜔=𝑣𝑟
Example:
Estimate the angular velocity of the Earth about the Sun. Note: it takes 8 minutes for the light from the Sun to reach Earth.
Answer:
Putting the value in formulas the equation
𝜔=2×10−7rads−1
Kinematic Equations With Mass
Define: Force is equal to mass time acceleration. F=ma
We know that
D= V_{t }+ 1/2 (at^{2})
And acceleration a=F/m put a value in Displacement formula
D= Vt + 1/2 (F/m) t^{2}
Mass= m = (Vt + 1/2 F t^{2})/D
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