**1-D MOTION**

**2-D MOTION OR MOTION IN A PLANE ****Þ**** **

Motion of any object in any of two axis involve (xy or yz or zx)

“Projectile Motion may be 2-D or even 3-D also but in general it is use to be 2-D Motion”

**-D PROJECTILE MOTION**

**VERTICAL MIRROR **

Vertical Mirror ® Gravity acts in vertical direction so we can use equations of motion

Time taken by the image of ball going up = time taken going down = t (Let)

Use v = u – g t

O = u sinq - g t

Total time

Now to get Maximum height

Using *v*^{2} = u^{2 }– 2g h

(O)^{2} = (usinq)^{2} – 2g H

**HORIZONTAL MIRROR**

Velocity remains constant so we can not use equation of Motion

**Note:** Untill any external reason present to change ucosq, like air flow ucosq remains Constant

Now ucosq = Const

R = (u cosq) T

Now Sin2q = 2 sinq Cosq

Now we have

**CONDITION **

“These three results only when can be use if initial point of projection and final point of projection are at same level.

**Example**: A body of mass m is projected upward with initial velocity then find time of flight, Maximum height attained and range attained by the body (g = 10m/s^{2})

**Solution **

**Maximum range: **To get maximum horizontal distance covered by any mass in projectile motion we must have a unique specified angle.

(Sin2q)_{max} = 1

2q = 90°

q = 45°

To get maximum range angle of projection should be 45 °,

**Note**: Here we are neglecting the effect of air resistance. If we Consider air resistance then this angle q should be little bit less than 45°

**SAME RANGE**

Mathematically there must be two different angle of projection for which we will get same range

Ball 1

Ball 2

Now if we assume

a + b = 90°

then b = 90° - a

If sum of angle of projection is = 90° and initial speed is same then both balls will have same range.

Note: Here in this Case only range will be same not time of flight and maximum height

R_{1} = R_{2}

T_{1} ¹ T_{2}

H_{1} ¹ H_{2}

In case of same range relation of time of flights

Here we know a + b = 90°

Ball 1

Ball 2

Now

In case of same range relation of maximum heights

Here again a +b = 90°

Ball 1

Ball 2

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