# Instantaneous velocity, speed and acceleration

## 4. Instantaneous velocity and speed

• Velocity of particle at any instant of time or at any point of its path is called instantaneous velocity.
• Again consider the graph 5b and imagine second point Q being taken more and more closer to point P then calculate the average velocity over such short displacement and time interval.
• Instantaneous velocity can be defined as limiting value of average velocity when second point comes closer and closer to the first point.
• Limiting value of Δx/ Δt as Δt approaches zero is written as dx/dt, and is known as instantaneous velocity.Thus instantaneous velocity is

• As point Q approaches point P in figure 5a in this limit slope of the chord PQ becomes equal to the slope of tangent to the curve at point P.
• Thus we can say that instantaneous velocity at any point of a coordinate time graph is equal to the slope of the tangent to the graph at that point.
• Instantaneous speed or speed is the magnitude of the instantaneous velocity unlike the case of average velocity and average speed where average speed over an finite interval of time may be greater than or equal to average velocity.
• Unit of average velocity , average speed, instantaneous velocity and instantaneous speed is ms-1 in SI system of units.
• Some other units of velocity are ft.s-1 , cm.s-1 .

## 5. Acceleration

• Acceleration is the rate of change of velocity with time.
• For describing average acceleration we first consider the motion of an object along X-axis.
• Suppose at time t1 object is at point P moving with velocity v1 and at time t2 it is at point Q and has velocity v2. Now average acceleration of object in moving from P to Q is

which is the change in velocity of object with the passage of time.
• Instantaneous acceleration can be defined in the same way as instantaneous velocity

• The instantaneous acceleration at any instant is the slope of v-t graph at that instant.

• In figure 7 instantaneous acceleration at point P is equal to the slope of tangent at this point P.
• Since velocity of a moving object has both magnitude and direction likewise acceleration depending on velocity has both magnitude and direction and hence acceleration is a vector quantity.
• Acceleration can also be positive, negative and zero.
• SI unit of acceleration is ms-2

### Acceleration Concept Map

Question:
A car moves in a straight road. The velocity at point A is 50 km/hr, It presses the accelerator paddle and reach point B in 10 sec. The velocity ar point B is 100Km/hr
Find the average acceleration of the car from point A to point B?
Solution
Initial Velocity =50km/hr= 13.88 m/s
Final Velocity = 100Km/hr= 27.77 m/s
Average Acceleration= Change in velocity/time taken= (27.77-18.88)/10=1.389 m/s/2
Question:
True and False Statement
a) Acceleration is a scalar quantity?
b) if the velocity of the object is zero,then acceleration will always be zero?
c) Instantaneous speed is always equal to magnitude of instantaous velocity
Solution
a) False
b) False
c) True

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