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Topics Covered

• Average Velocity
• Average Speed
• Instantaneous Speed
• Instantaneous Velocity
• Acceleration

Average Velocity & Speed

`\color{fuchsia} ★ \color{fuchsia} ul{\mathbf( "Average Velocity")} `

`color(green)("Average velocity")` is defined as the change in position or displacement (Δx) divided by the time intervals (Δt), in which the displacement occurs :

`color(blue){text(velocity) = barv_(avg)=(x_2-x_1)/(t_2-t_1)=(Deltax)/(Deltat)]`

where `color(blue)(x_2)` and `color(blue)(x_1)` are the positions of the object at time `color(blue)(t_2)` and `color(blue)(t_1)`, respectively.

The `ul"SI unit for velocity"` is `color(green)(m//s)` or `color(green)(m s^(–1))`.

`\color{green} ✍️` Like displacement, average velocity is also a vector quantity.

`\color{green} ✍️` The average velocity can be positive or negative depending upon the sign of the displacement.

`\color{green} ✍️` But for one dimensional motion direection can be taken are of + or - sign.

`\color{green} ✍️` average Velocity is zero if the displacement is zero.

Fig. shows the x-t graphs for an object, moving with positive velocity (Fig. a), moving with negative velocity (Fig. b) and at rest (Fig. c).

`\color{fuchsia} ★ \color{fuchsia} ul{\mathbf( "Average Speed")} `

`color(green)("Average speed")` is defined as the total path length travelled divided by the total time interval during which the motion has taken place.

`color(blue)(text(Average speed) = text(Total path length) / text(Total time interval))`

Average speed has obviously the same unit (`m s^(–1)`) as that of velocity. But it does not tell us in what direction an object is moving.

`\color{green} ✍️` Speed is always positive.

`\color{green} ✍️` If the motion of an object is along a straight line and in the same direction, the magnitude of displacement is equal to the total path length. In that case, the magnitude of average velocity is equal to the average speed.
Q 3083256147

A car is moving along a straight line, say `OP` in Fig. It moves from `O` to `P` in `18 s` and returns from `P` to `Q` in `6.0 s`. What are the average velocity and average speed of the car in going (a) from `O` to `P` ? and `(b)` from `O` to `P` and back to `Q ?`


(a) `text(Average velocity) = text(Displacement)/text(Time interval)`

`barv = (+360 m)/(18 s) = +20 ms^(-1)`

`text(Average speed) = text(Path length)/text(Time interval)`

` = (360 m)/(18 s) = 20 m s^(-1)`

Thus, in this case the average speed is equal to the magnitude of the average velocity.

(b) In this case,

`text(Average velocity) = text(Displacement)/text(Time interval) = (+240 m)/((18+6.0)s)`

` = +10 ms^(-1)`

`text(Average speed) = text(Path length)/text(Time interval) = (OP+PQ)/(Deltat)`

` = ((360+120 )m)/(24 s) = 20 ms^(-1)`

Instantaneous Velocity & Speed

`\color{fuchsia} ★ \color{fuchsia} ul{\mathbf( "Instantaneous Velocity")} `

`\color{green} ✍️` The velocity at an instant is defined as the limit of the average velocity as the time interval `Δt` becomes infinitesimally small. In other words,


We can use this equation for obtaining the value of velocity at an instant.

`\color{fuchsia} ★ \color{fuchsia} ul{\mathbf( "Instantaneous Speed")} `

`\color{green} ✍️` Instantaneous speed or simply speed is the magnitude of velocity. For example, a velocity of `+ 24.0 m s^(–1)` and a velocity of `– 24.0 ms^(–1)` — both have an associated speed of `24.0 m s^(-1)`.

Q 3013456340

The position of an object moving along x-axis is given by `x = a + bt^2` where `a = 8.5` m, b = 2.5 m `s^(–2)` and t is measured in seconds. What is its velocity at `t = 0 s` and `t = 2.0 s.` What is the average velocity between `t = 2.0 s` and `t = 4.0 s` ?


In notation of differential calculus, the velocity is `v = (dx)/(dt) = d/(dt) (a+b t^2) = 2 bt = 5.0 t ms^(-1)`

At `t = 0 s , v = 0 ms^(-1)` and `t = 2.0 s`

`v = 10 ms^(-1)`

`text(Average velocity) = ( x (4.0) - x(2.0))/(4.0-2.0)`

` = (a+16 b - a-4b)/(2.0) = 6.0xx b`

`= 6.0 × 2.5 =15 m s^(-1)`


`color(green)(ul"Acceleration")` is the rate of change of velocity of an object with respect to time.

`\color{fuchsia} ★ \color{fuchsia} ul{\mathbf( "Average Acceleration")} `

The average acceleration `bara` over a time interval is defined as the change of velocity divided by the time interval.


where `color(blue)(v_2)` and `color(blue)(v_1)` are the instantaneous velocities or simply velocities at time `color(blue)(t_2)` and `color(blue)(t_1)`.

`\color{green} ✍️` It is the average change of velocity per unit time.

`\color{green} ✍️` The SI unit of acceleration is `m s^(–2)`.

`\color{green} ✍️` On a plot of velocity versus time, the average acceleration is the slope of the straight line connecting the points corresponding to `(v_2, t_2)` and `(v_1, t_1)`.

`\color{fuchsia} ★ \color{fuchsia} ul{\mathbf( "Instantaneous Acceleration")} `

`color(green)("Instantaneous acceleration")` is defined in the same way as the instantaneous velocity.


The acceleration at an instant is the slope of the tangent to the `v–t` curve at that instant.

`\color{green} ✍️` Like velocity, acceleration can also be positive, negative or zero.

`\color{green} ✍️` Acceleration may result from a change in speed (magnitude), a change in direction or changes in both.

Position-Time Graph

Position-time graphs for motion with positive, negative and zero acceleration are shown in Figs. (a), (b) and (c), respectively.

`\color{green} ✍️ \color{green} \ul mathbf(KEY \ CONCEPT)` Note that the graph curves

`color(blue)(=>"Upward for positive acceleration")`;

`color(blue)(=>"Downward for negative acceleration")` and

`color(blue)(=>"A straight line for zero acceleration").`

Velocity-Time Graph

Fig. shows velocity time graph for motion with constant acceleration for the following cases :

`(a)` An object is moving in a positive direction with a positive acceleration.

`(b)` An object is moving in positive direction with a negative acceleration.

`(c)` An object is moving in negative direction with a negative acceleration.

`(d)` An object is moving in positive direction till time `t_1`, and then turns back with the same negative acceleration.

`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)`

An interesting feature of `color(green)(ul"a velocity-time graph for any moving object")` is that `color(blue)(ul"the area under the curve represents the displacement over a given time interval.")`