Motion in Plane

Velocity

We know that the average velocity of an object is the ratio of displacement and corresponding time interval. So, average velocity can be given by following equation:

`v=(Δr)/(Δt)`

`=(Δx\i + Δyj)/(Δt)=i(Δx)/(Δt)+j(Δy)/(Δt)`

Or, `v=v_x\i +v_yj`

Velocity (instantaneous velocity) is given by the limiting value of the average of velocity as the tie interval approaches zero:

We can express v in component form as follows:

`v=(dr)/(dt)`

Or, `v=i(dx)/(dt)+j(dy)/(dt)=v_x\i+v_y\j`

Where `v_x=(dx)/(dt)` and `v_y=(dy)/(dt)`

If expressions for coordinates x and y are known as functions of time, these equations can be used to find v_x and v_y

Value of v can be given by following equation:

`v=sqrt(v_x^2+v_y^2)`

Direction of v is given by angle θ

tan θ `=(v_y)/(v_x)`

Or, `θ=ta\n^(-1)((v_y)/(v_x))`

Acceleration

We know that average acceleration is given by change in velocity divided by time interval. So, it can be given by following equation:

`a=(Δv)/(Δt)=(Δ(v_xi+v_yj))/(Δt)`

`=(Δv_x)/(Δt)i+(Δv_y)/(Δt)j`

Or, `a=a_x\i+a_y\j`

Acceleration (instantaneous acceleration) is the limiting value of average acceleration as the time interval approaches zero. This can be given by following equation.

Or, `a=a_xi+a_yj`

Where, `a_x=(dv_x)/(dt)` and `a_y=(dv_y)/(dt)`

Motion in a Plane with Constant Acceleration

Let us assume that an object is moving in x-y plane with constant acceleration a. After some time, the average acceleration will become equal to this constant value. If velocity is v₀ at time t = 0 and v at time t, then

`a=(v-v_0)/(t-0)=(v-v_0)/t`

Or, `at=v-v_0`

Or, `v=v_0+at`

In terms of components:

`v_x=v_(0x)+a_x\t`

`v_y=v_(0y)+a_y\t`

Change of Position: Let us assume that r₀ and r be the position vectors of particle at time 0 and t and velocities at these instants are v₀ and v respectively. The average velocity over this time interval t can be given as follows:

`v_(av)=(v_0+v)/2`

Now, displacement can be given by following equation:

`r-r_0=((v+v_0)/2)t`

`=(((v_0+at)+v_0)/2)t`

`=v_0t+1/2at^2`

Or, `r=r_0+v_0t+1/2at^2`

In component form, above equation can be written as follows:

`x=x_0+v_(0x)t+1/2a_x\t^2`

`y=y_0+v_(0y)t+1/2a_y\t^2`

It can be said that motion in a two-dimensional plane can be treated as two separate simultaneous one-dimensional motions with constant acceleration along two perpendicular directions.

Relative Velocity in Two Dimensions

If A and B are moving with velocities v_A and v_B, then velocity of A relative to B is:

`v_(AB)=v_A-v_B`

Similarly, velocity of object B relative to A is:

`v_(BA)=v_B-v_A`

So, `v_(AB)=v_(BA)`

And `|v_(AB)|=|v_(BA)|`