# Motion in a Plane with Constant Acceleration

The average velocity of the object in the time interval between t and tâ€™ is given by= â€¦.(4.1)

If =are velocities at times t and tâ€™ respectively, then constant acceleration of the motion of the object is given by

= â€¦..(4.2)

Suppose that at t = 0, the velocity of the object is . If at time t = 0, u

_{x}and u

_{y}are component of the velocity of the object along X-axis and Y-axis respectively, then

= u

_{x}+ u

_{y }â€¦.. (4.3)

= â€¦..(4.4)

and = â€¦..(4.5)

=

= +(tâ€™ â€“ t) â€¦(4.6)

or

the above equation can also be obtained from equations (4.2) by expanding it.

If a

_{x}and a

_{y}are magnitudes of the components of acceleration along X-axis and Y-axis respectively then

= a

_{x}+ a

_{y}â€¦..(4.7)

such that

= (u

_{x}+ u

_{y}) + (a

_{x}+ a

_{y}) t

_{x}and v

_{y}are magnitudes of the component of velocity of time t, then

v

_{x}+ v

_{y}= (u

_{x}+ u

_{y}) + (a

_{x}+ a

_{y}) t

or v

_{x}+ v

_{y}= (u

_{x}+ a

_{x}t) + (u

_{x}+ a

_{y}t)

v

_{x }= u

_{x}+ a

_{x}t â€¦ (4.8)

and v

_{y }= u

_{y}+ a

_{y}t â€¦ (4.9)

= (tâ€™ â€“ t) â€¦.(4.10)

Also, =

= (tâ€™ â€“ t)

= â€¦..(4.11)

=

x+ y = (x

_{0}+ y

_{0}) + (u

_{x}+ u

_{y})t + ) t

^{2}

or x+ y = (x

_{0}+ u

_{x}t+ + (y

_{0}+ u

_{y}t +

x = x

_{0}=u

_{x}t+ â€¦(4.13)

and y = y

_{0}=u

_{y}t+ â€¦(4.14)

_{0}, y

_{0}, u

_{x}, u

_{y}, a

_{x}and a

_{y}.

Let us consider a particular case of throwing an object in a vertical XY-plane with some velocity making an angle Î¸ with horizontal from point say P(x

_{0}, y

_{0}) in the plane. In such a case, a

_{x}= 0 and a

_{y}= -9.8 ms

^{-2}. For certain values of u

_{x}and u

_{y}, the values of x and y at time t = 1s, 2s, 3s,â€¦ can be found. If we plot a graph between x and y, the graph will be a parabola as shown in the figure.