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Chapter 7: Motion in 2 Dimensions

 When an object is projected so that it moves horizontally as well as vertically, it is said to be a 2-

 D projectile.

 When we solve such problems, we use the same equations of motion to do calculations in the

horizontal and vertical directions independently.

Note: horizontally, ; vertically,  

Horizontally Launched Projectile

Example 1: A cannonball fired horizontally on the top of a 75 m high cliff moves with an initial

speed of 1769 ft/s (note: 1 ft = 0.3048 m).

a)  Determine how long it takes for the cannonball to drop into the sea.

 b)  Determine the horizontal displacement, d  x .

c)  Determine the final velocity of the cannonball just before hitting the water.

Solution:

a)   Vertically <y>        

 

 

 

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 b)  Horizontally <x>

 

   

 

 

     

c)   Vertically <y>            

  √   

 

( )  The final velocity,  , is , below the horizontal.

Circular Motion

To change an object’s velocity, we need to apply a non-zero net force ( ). When an

object moves in a curve, the direction of the object constantly changes and so its velocity is

changing (even if its speed remains constant). Thus a force is needed to make an object move in

a curved motion.

In circular motion, the acceleration of an object moving with constant speed, v, with a radius of curvature, r, is given by:

 

 where ac is the centripetal acceleration.

θ   

V 

 

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The net force for uniform circular motion is . F c and ac are always directed towards the

centre of the motion.

Example: The Earth revolves around the sun in about 365 days in a year. Calculate the

centripetal force acting on the Earth if we assume uniform circular motion.

Data:

mass of Earth = 5.98 × 1024 kg

radius of Earth’s orbit around the Sun = 1.50 × 1011 m

 

 

 

This centripetal force on the Earth is caused by the gravitational pull of the Sun on the Earth.

Forces in 2-D: Inclined Plane

 At the point of moving,

 

and

 

(on a level surface)

 

Moving at a constant

speed,

 

Find by drawing a graph of  or against or :

 

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Now find on an inclined plane:

Case A: Frictionless

 

Case B: Not Frictionless

 At the point of moving, :

 

SOH

CAH

TOA 

8 data points

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Example

Calculate the acceleration down the slope when

a)  the ramp is frictionless

 

 

 

 b)  the ramp has a coefficient of friction (kinetic) of 0.05 with the object’s base surface