Vectors 1

Adding Vectors

• Adding two or more vectors you are finding the resultant force
• Example: Jemaima goes for a walk. She walks 3m North and 4m East. She has walked 7m but isnt 7m from her starting point. find the magnitude and direction of her displacement. Draw the vector, drawing a line from the tip of one to the tail of the other to find the resultant giving you your magnitude. Using Phythagoras R=(3^2)+(4^2)= 5m, to find the bearing of the new position using trig you know the opp and the adj meaning tan0=4/3 0=53.1 R=5m at 53.1 

• Remember SOH CAH TOA!   

• It is useful to split the vectors in to horizontal and vertical components when you have the resultant forces
• The vertical component Vy ... sin0=Vy/V... Vy=Vsin0.
• The horizontal component Vx ... cos0=Vx/V... Vx=Vcos0.
• Resolving vectors is useful because the two components of a vector don't affect eachother meaning you can deal with the two directions separately.
• Example; Charley's amazing floating home is travelling at a speed of 5 ms^-1 at an angle of 60 up from the horizontal. Find the vertical and horizontal components. The hoizontal component is 5cos60=2.5ms^-1, The veritacally component is 5sin60=4.33ms^-1

Equipment:Iron ball, Start switch, electronic timer, timing gate, solenoid. 

Method: The iron ball is released by operating the start switch, which also starts the timer. The time taken for the ball to fall vertical distance s is measured as it passes through the timing gate.

Calulation: Since S=Ut+0.5at^2 and U=0 we can write S=o.5gt^2. This equation can be used to calculated g directly from one measurement of the falling ball. However, it is better to take several readings of the time to fall, take the most consistent ones, average these and then use the avarage to calculate g, since this should lead to a more reliable result. Even better is to notice that s is proptional to t^2. If we vary S, and plot the values of t^2 we obtain against S, the slope of the gradient will be S=0.5g. From the formula y=mx+c, s=y, o.5g=mx, t^2=c.

• Aristole reckond that if two objects of different masses were dropped from the same hieght, the heavier object would always hit the ground frist.
• Galileo disagreed, he thought that all objects should acclerate towards the ground at the same rate maning the objects should hit the ground at the same time and the only reason this wouldnt occur would be because of air resistance.
• Gailileo thoery overturned Aristole thoery 

• Defined as "The motion of an object undergoing an acceleration of 'g.'"
• Accleration is a vector quantity 
• Unless you're given a different value, take the magnitude of g as 9.81ms^-1 though it varies slightly at different points on the Earth's surface.
• The only force acting on an object in free fall is its weight.
• Objects can have an intitial velocity in any direction and still undergo free fall as long as the force providing the initial velocity is no longer acting.
• g is always downwards and negative
• U and V can be positive or negative
• t is always positive
• S can be negative or positive.

Example: Sharon fires a scale model of a tv talent show presenter^'1 from 1.5m. How long does it take to hit the ground, and how far does it travel? Assume the model acts as a particle, the ground is horizontal and there is no air resistance. 7

Think about the vertical motion first: 

• It's constant acceration under gravity...
• You know u=0 (no velocity at first), s=-1.55, g=-9.81. t=?
• Using s=0.5gt^2 rearragne to t=2s/g= 2x-1.5/-9.81=0.55s
• So the model hits the grounf after 0.55 seconds. 

Then do the horizontal motion:

• The horizontal motion isn't affected by gravity or any other force, so it moves at a constant speed.
• That means you can just use good old speed = distance/ time.
• Now Vh=100ms^-1, t=0.55s and a=0
• Sh=Vht=100x0.55=55m

Method for an angle:

• Resolve the initial velocity into horizontal and vertical components.
• Use the vertical component to work out how long it's in the air and how high it goes.
• Use the horizontal component to work out how far it goes while it's in the air.