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Velocity And its types ,Acceleration
Distance & Displacement:
Distance and displacement are two quantities that may seem to mean the same thing yet have distinctly different definitions and meanings.
Distance is a scalar quantity that refers to “how much ground an object has covered” during its motion.
Displacement is a vector quantity that refers to “how far out of place an object is”; it is the object’s overall change in position.
Units of Distance & Displacement: meter is the S.I. units of distance & displacement.
In the figure:1, The distance is indicated by dotted lines from A to B. That is entire path travelled by the body to reach the destination ‘B’.
That total path travelled by the body is called Distance . And if a straight line drawn from point A to Point B, then that is the possible way to reach point B in shortest path. That shortest path is called as Displacement
Some more examples:
Differences between Distance & Displacement:
Distance & Displacement in different cases:
Case 1:
Motion in straight path:
Displacement is the shortest distance between two points, while distance is the length of the path travelled. Therefore, if the body takes the shortest path between two points, distance and displacement will be equal. Imagine it in this way: Between two points you can draw any line-curved, with turns, etc, but you can draw only one straight line connecting the points which will be of the shortest distance. If an object travels across the curved line, the distance is equal to the length of the curved line, while displacement is equal to the length of the straight line. In case the object travels in this straight line, distance will be equal to the length of this line, which is also equal to displacement.
When an object is moving in straight path, then distance & displacement can be measured in same way because entire path travelled by the object & shortest path travelled by the object is same, so distance is equals to displacement.
Special case:
Here, displacement can be negative in some cases. To understand this, we need to take a fixed reference point for the motion of a body.
Case 2:
Motion in curved path:
When an object is moving in curved path, then distance & displacement can never be measured in same way because entire path travelled by the object & shortest path travelled by the object can never be same, so distance is not equal to displacement.
Case 3:
Motion in circular path:
If a body is moving in circular path, then the body will have distance & displacement also. But the displacement will be zero if the body comes back to the same point again in its path. In this giant wheel, there are 20 cabins situated at different places & each cabin is fixes at 1m apart from each other. Then if the giant wheel started moving then, Cabin-1 starts moving from its position (Reference point), if it moves to next position, then distance & displacement will not be same, because it is not moving in straight path. If the Cabin-1 again comes back to same reference point, then displacement can be considered as zero
Speed:
Speed is the distance travelled per unit of time. It is how fast an object is moving. Speed is the scalar quantity that is the magnitude of the velocity vector. It doesn’t have a direction. Higher speed means an object is moving faster. Lower speed means it is moving slower. If it isn’t moving at all, it has zero speed.
The most common way to calculate the constant velocity of an object moving in a straight line is the formula:
S = d / t
where,
S is the speed (sometimes denoted as v, for velocity)
d is the distance moved
t is the time it takes to complete the movement.
Units for Speed:
The SI units for speed are m/s (metre per second). In everyday usage, kilometre per hour or miles per hour are the common units of speed. At sea, knots (or nautical miles per hour) is a common speed.
Types of speed:
There are four types of speed and they are:
1. Uniform speed 2. Non-Uniform speed
3. Average speed 4. Instantaneous speed
Uniform speed:
An object is said to be in uniform speed when the object covers equal distance in equal time intervals.
Ex: Movement of blades of a ceiling fan.
Non-Uniform speed:
An object is said to be in non-uniform speed when the object covers a different distance in equal intervals of times.
Ex: The motion of a train.
Average speed:
Average speed is defined as the uniform speed which is given by the ratio of total distance travelled by an object to the total time taken by the object.
The average speed of a body in a certain time interval is the distance covered by the body in that time interval divided by time. So, if a particle covers a certain distance s in a time t1 to t2, then the average speed of the body is:
\({{v}_{avg}}=\frac{S}{{{t}_{2}}-{{t}_{1}}}=\frac{Total\,\,dis\tan ce}{Total\,\,time}\)
Now let us look at some of the examples to understand this concept easily
Instantaneous speed:
When an object is moving with variable speed, then the speed of that object at any instant of time is known as instantaneous speed.
The average velocity tells us how fast an object has been moving over a given time interval but does not tell us how fast it moves at different instants of time during that interval. For this, we define instantaneous speed. It is the rate of change of distance with respect to time.
Velocity:
Velocity is defined as a rate of change of displacement. In a simple, velocity specifies speed of body and its direction. The speed of a car traveling north on a major freeway and the speed of a rocket launching into space can both be measured using velocity.
Velocity Formula:
The most common way to calculate the constant velocity of an object moving in a straight
line is with this formula: \(V ={d\over t}\)
V is the velocity
d is the displacement
t is the time it takes to complete the movement
Units of Velocity:
The SI (international) units for velocity are m/s (metre per second), but velocity may also be expressed in any units of distance per time. Other units include miles per hour (mph), kilometre per hour (kph), and kilometre per second (km/s).
1 km/hr = 5/18 m-s-1
Types of Velocity:
T here are four types of speed and they are:
1. Uniform velocity 2. Non-Uniform velocity
3. Average velocity 4. Instantaneous velocity
Uniform velocity:
It is the condition in which a body covers equal distance in equal interval of the time in a specified direction.
Ex: The rotation of Moon around the earth etc.
Non-Uniform Velocity:
When a body covers unequal distances in equal intervals of time in a specified direction or equal distances in equal intervals of time, but its direction changes then body is said to be moving with non-uniform velocity.
Ex: A car moving on the road etc.
Average velocity:
The average velocity of a body in a certain time interval is given as the displacement of the body in that time interval divided by time. So, if a particle covers a certain displacement A’!B in a time t1 to t2, then the average velocity of the particle is:
\({{v}_{avg}}=\frac{S}{{{t}_{2}}-{{t}_{1}}}=\frac{Total\,\,dis\tan ce}{Total\,\,time}\)
Instantaneous velocity:
In simple words, the velocity of an object at that instant of time. Instantaneous velocity definition is given as ”The velocity of an object under motion at a specific point of time.”
If the object possesses uniform velocity then the instantaneous velocity may be the same as its standard velocity.
Acceleration:
Acceleration is defined as the rate of change of velocity with respect to time. Acceleration is a vector quantity as it has both magnitude as well as direction. When the velocity of an object changes it is said to be accelerating. Basically, acceleration is the word which is often used to describe a state of increasing speed.
A change in the direction of motion results in an acceleration even if the moving object neither sped up nor slowed down. That’s because acceleration depends on the change in velocity
\(Acceleration(a)=\frac{change\,\,in\,\,velocity}{time}=\frac{{{V}_{f}}-{{V}_{i}}}{t}\)
Vi = initial velocity, Vf = Final velocity, t= time
Types of acceleration:
There are different types of accelerations are there. Out of those, some of the types we will discuss here.
1. Positive Acceleration 2. Negative Acceleration
3. Zero Acceleration 4.Uniform Acceleration
5. Non-Uniform Acceleration 6. Average Acceleration
7Instantaneous acceleration
Positive Acceleration:
When the motion of the body is in same direction as that of acceleration, we say the acceleration is positive. In another terms, if initial velocity (U) is less than final velocity (V), then it is called as positive acceleration. (U
Negative Acceleration:
When the motion of the body is opposite to the direction of acceleration, we say the acceleration is negative. In another terms, if initial velocity (U) is greater than final velocity (V), then it is called as positive acceleration. (U>V)
Ex: A bus or car moving with increasing speed, then if we apply the break, the speed of the car decreases. Then it has negative acceleration. So we can say that it is decelerating.
Zero Acceleration:
If the acceleration is zero, it means that the velocity is steady, not increasing or decreasing in magnitude. If the object has zero velocity and zero acceleration, it is stationary and not changing that stationary condition.
Ex: An apple thrown in space, in space that apple can’t change its velocity, so no acceleration exists for it.
Constant acceleration means that the acceleration does not change. If the acceleration is zero then the acceleration does not change, so we can say that the body is moving with uniform velocity.
Uniform Acceleration & Non-Uniform Acceleration:
If an object’s speed (velocity) is increasing at a constant rate then we say it has uniform acceleration. The rate of acceleration is constant. This acceleration is the same over time. If a car speeds up then slows down then speeds up it doesn’t have uniform acceleration. This is called non-uniform acceleration.
“Uniform acceleration is equal change of equal velocity in equal intervals of time.” “Non-Uniform acceleration is unequal change of non-equal velocity in equal intervals of time.”
Average Acceleration:
Acceleration is the rate of change for velocity, that is, change in velocity over a specified period of time. Average acceleration is the final velocity minus the initial velocity per time taken.
Average Acceleration \({{A}_{avg}}=\frac{\Delta V}{\Delta t}=\frac{change\,\,in\,\,Velocity}{change\,\,in\,\,time}=\frac{{{V}_{f}}-{{V}_{i}}}{{{t}_{f}}-{{t}_{i}}}\)
Instantaneous Acceleration:
When an object is moving with variable acceleration, then acceleration of the object at any instant of time is called instantaneous acceleration. It may be also defined as the limiting value of average acceleration over an interval of time, which approaches to zero.
In the above diagram, car have zero initial velocity, after that car started moving. At different instants of time, different accelerations are mentioned in the above diagram. This above diagram shows that non-uniform accelerated motion also.
Acceleration due to gravity(g):
When the object falls towards the earth due to the earth’s gravitational force, it something we call as free fall of the object. So, during the free fall, the only force acting on the object is the gravitational force of the earth. The acceleration due to gravity is the acceleration produced in the freely falling body due to the influence of the gravitational pull of the earth.
Acceleration due to gravity is the acceleration gained by an object due to the gravitational force. Its SI unit is m/s2.
It has both magnitude and direction, hence, it’s a vector quantity. Acceleration due to gravity is represented by g. The standard value of g on the surface of the earth at sea level is 9.8 m/s2, but its values vary. Like, for example, the acceleration due to gravity on the moon is different from that of the earth.
Usually we consider the values for g = 9.8 m/s2 (or) 10 m/s2WORKED EXAMPLES
Example 1:
An object moves, along a line, from point A to B to C and then back to B again as shown in
the figure below.
a) Find the distance covered by the moving object.
b) Find the magnitude of the displacement of the object
Solution:
a) The total distance d covered by the object is
d = AB + BC + CB = 5 km + 4 km + 4 km = 13 km
b) The magnitude of the displacement is equal to the distance from A (initial position) to B (final position) which is equal to 5 km.
Example 2:
A runner travels around rectangle track with length = 50 meters and width = 20 meters. After he runs along the rectangle tract twice, hr returned to starting point. Determine distance and displacement.
Solution:
Circumference of rectangle = 2(50) m + 2(20) m = 100 m + 40 m = 140 meters.
Travels around rectangle 2 times = 2(140 meters) = 280 meters.
Distance = 280 m.
Displacement = 0 m. (the runner returns to the starting point)
Example 3:
In travelling from Pune to Nagpur, Rahul drove his bike for 2 hrs at 60 kmph, 3 hrs at 70 kmph.
Solution:
We know that, Distance = Speed × Time
So, in 2 hours, distance covered = 2 × 60 = 120 km
in the next 3 hours, distance covered = 3 × 70 = 210 km
Total distance covered = 120 + 210 = 330 km
Total time = 2 + 3 = 5 hrs
Average Speed = \(=\frac{total\,\,dis\tan ce}{total\,time\,\,taken}=\frac{330}{5}=66kmph\)
Example 4:
Calculate the average velocity at a particular time interval of a person if he moves 7 m in 4 sec and 18 m in 6 sec along x-axis?
Solution:
Initial distance traveled by the person Xi = 7 m,
Final distance traveled Xf = 18 m,
Initial time interval Ti = 4 sec
Final time interval Tf = 6 sec
Average velocity \({{V}_{avg}}=\frac{{{X}_{f}}-{{X}_{i}}}{{{T}_{f}}-{{T}_{i}}}=\frac{18-7}{6-4}=5.5m/\sec .\)
Example 5:
If a body stated moving with initial velocity 5 m/sec and travelled 10 sec and stopped at some point with final velocity of 35 m/sec. then what is acceleration?
Solution:
Initial velocity = 5 m/sec
Final velocity = 35 m/sec
Time = 10 sec
Acceleration (a) =
\(\frac{change\,\,in\,\,\,Velocity}{time}=\frac{{{V}_{f}}-{{V}_{i}}}{t}=\frac{35-5}{10}=3m\,/{{\sec }^{2}}\)
Example 6:
Your friend’s new car can go from 0 to 60 m/s in 7 sec. What is the acceleration?
Solution:
The final velocity, Vf = 60 m/s; The initial velocity, Vi = 0 m/s;
The final time, Tf = 7 sec; The initial time, Ti = 0 sec;
\({{A}_{avg}}=\frac{\Delta V}{\Delta t}=\left( 60m/s \right)/7s=12.86m/{{s}^{2}}\)
Example 7:
When driving along a side-road at 45 m/s, you see a child run out into the street, and rapidly slow to 3 m/s, within 1.5 sec. What is your acceleration?
Solution:
The final velocity, Vf = 3 m/s; the initial velocity, Vi = 45 m/s; and the time, t = 1.5 sec.
\({{A}_{avg}}=\frac{\Delta V}{\Delta t}\) = (3 m/s - 45 m/s) / 1.5 sec = (- 42 m/s) / 1.5 sec = - 28 m/s2
Notice, the negative sign, indicating the object slowed down.
Example 8:
A bus accelerates with an initial velocity of 10 m/s for 5s then 20m/s for 4s finally for 15 m/s for 8s. What can be said about the average acceleration of the bus?
Solution:
It is given that, the velocities of the bus at different time intervals is,
V1 = 10 m/s, V2 = 20m/s, V3 = 15m/s
The time intervals for which the object possesses these velocities are
t1 = 5s, t2 = 4s, t3 = 8s
Hence, over the interval, the total velocity can be given as the sum of these velocities.
v=10+20+15=45m/s
Similarly, the total time interval can be given as the sum of these intervals,
t=t1+t2+t3=5+4+8=17s
Using the above formula for average acceleration, we get,
Average Acceleration=v/t
Average Acceleration=45/17=2.65ms-2