Scalars and Vectors

Scalars only have size, but vectors have size and direction. A scalar only has a magnitude, whilst a vector has a magnitude and direction. Force and velocity are both vectors. Some scalar quantities are mass, temperature, time, length, speed, and energy. Some vector quantities are displacement, force, velocity, acceleration, and momentum. Adding vectors involves pythagoras and trigonometry. Adding two vectors is called finding the resultant of them. To find the magnitude of two vectors at right angles to one another, you use pythagoras to work out the missing side. To work out the angle, you use inverse tan of the opposite side divided by the adjacent side. A good understanding of trigonometry is useful in mechanics. Use the same method for resultant forces or velocities. Start by drawing a diagram of what you know. If the vectors are the same magnitude the angle is 45 degrees. It's useful to split a vector into horizontal and vertical components. This is the opposite of finding the resultant - you take a vector and split it into a horizontal and vertical component. The horizontal component is given by v(horizontal)=v cos (theta), where theta is the angle. The vertical component is given by v(vertical)=v sin theta. v is the inital vector. Resolving is useful because the two components of a vector don't affect each other. This means you can deal with the two directions completely seperately. Only the vertical component is affected by gravity.

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Motion with Constant Acceleration

Uniform acceleration is constant acceleration. There are four main equations that you use to solve problems involving uniform acceleration. Acceleration is the rate of change of velocity. From this definition you get a=(v-u) divided by t so v=u+at. u=initial velocity, v=final velocity, a=acceleration and t=time taken.  s=average velocity x time. If acceleration is constant, the average velocity is just the average of the initial and final velocities, so: s=(u+v) divided by 2 x time. s=displacement. Substitute the expression for v from equation 1 into equation 2 to get s=(u+u+at)xt divided by 2, which simplifies to s=ut+0.5 x a x t squared. Use equation 1 in the form a=(v-u) divided by t. Multiply both sides by s, where s=(u+v) divided by 2 x time. This gives as=(v-u) divided by t x (u+v)t divided by 2. 2as=v squared -uv+uv-u squared, so v squared=u squared+2as. When working with objects acting under gravity, take g=9.81 metres per second squared (unless otherwise stated) and g is negative in a downwards direction. To work out unknown quantities from a question write down what you know and use the appropiate formula to work out what you want to know. u=0 if the object starts from rest. You have to learn the constant acceleration equations. The questions using these equations are easy marks in the exam, so you'd be a bit stupid not to learn them... 

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Free Fall

Free fall is when there's only gravity and nothing else. Free fall is defined as "the motion of an object undergoing an acceleration of 'g'". Acceleration is a vector quantity, and g acts vertically downwards, with a magnitude of 9.81 metres per second squared unless otherwise stated. The only force acting on an object in free fall is its weight. Objects can have an initial velocity in any direction and still undergo free fall as long as the force providing the initial velocity is no longer acting. You can measure g by using an object in free fall. You need to be able to sketch a diagram of the apparatus, describe the method, list the measurements you take, explain how 'g' is calculated and be aware of sources of error. You need: a ball bearing attached to an electromagnet, a circuit with a switch and a trapdoor connected to a timer. Measure the height from the bottom of the ball bearing to the trapdoor. Flick the switch to release the ball bearing from the electromagnet and start the timer. The ball bearing falls and hits the trapdoor, breaking the circuit and stopping the timer. Use the time t measured by the timer and the height that the ball fell, using h=0.5 x g x t squared. The most significant source of error will be in the measurement of h. Using a ruler the uncertainty will be about 1mm. You can just replace a with g in the equations of motion. You need to be able to work out speeds, distances and times for objects in free fall. g is constant acceleration so you can use the constant acceleration equations. But g acts downwards so be careful about directions. g is usually negative, t is always positive, and u, v and s can be positive or negative. For a object that just falls, u=0 and a=g. For an object thrown in the air, a=g again, and the equations are just as normal. For an object thrown down you need to consider initial velocity, and distances will be negative.

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Free Fall and Projectile Motion

Aristotle - heavy objects fall faster than lighter objects. One of his famous theories was that if two different objects of different mass were dropped from the same height, the heavier object would hit the ground first. Galileo - all objects in  free fall accelerate uniformly. Galileo thought all objects accelerate towards the ground at the same rate - so objects with different weights should hit the ground at the same time. He also reckoned they didn't seem to do this because of the effect of air resistance on different objects. He did something called the inclined plane experiment - rolling balls down a slope. Galileo found that by rolling a ball down a smooth grove in an inclined plane, he reduced the effect of air resistance whilst slowing the ball's fall at the same time. He found by repeating the experiment that the distance the ball travelled was proportional to the square of the time taken. The ball was accelerating at a constant rate. Galileo tested his theories using experiments. His success was down to the systematic and rigorous experiments he used to test his theories. These experiments could be repeated and the results described mathematically and compared. You have to think of horizontal and vertical motion seperately. Think about vertical motion first - u=0, s=distance from ground (can be +ve or -ve) and a=g=9.81 metres per second squared. Use s=0.5xgxt squared. t= square root of 2s divided by g. Then do the horizontal motion: this moves at a constant speed (gravity doesn't affect it). So you can use speed=distance/time. It's slightly trickier if it starts off at an angle. Resolve the velocity into vertical and horizontal components. Use the vertical component to work out how long it is in the air and/or how high it goes. Use the horizontal component to work out how far it goes while in the air.

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Displacement-Time Graphs

Acceleration means a curved displacement-time graph. A graph of displacement (y-axis) against time (x-axis) for an accelerating object always produces a curve. You can plot a displacement-time graph once you have a relationship between s and t. E.g if a=2 metres per second squared and u=0 metres per second then using s=ut+0.5at squared and substuting in the values, you get s=t squared. Different accelerations have different gradients. If the object has a different acceleration it'll change the gradient of the curve. Bigger acceleration=steeper curve, higher gradient, smaller acceleration=shallower curve, less gradient. A deceleration of the same magnitude will be the graph flipped around and the curve gradually decreases to a horizontal line. The gradient of a displacement-time graph tells you the velocity. When the velocity is constant the graph's a straight line. Velocity =change in displacement/time taken. On the graph this is change in y/change in x, i.e the gradient. It's the same with curved graphs. If the gradient isn't constant it means the object is accelerating. To find the velocity at a certain point you need to draw a tangent to the curve at that point and find the gradient.

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Velocity-Time Graphs

Note: the difference between a speed-time and a velocity-time graph is that velocity-time graphs can have a -ve part to show something travelling in the opposite direction. The gradient of a velocity-time graph tells you the acceleration. Acceleration is change in velocity/time taken, so acceleration is just the gradient of a velocity-time graph. Uniform acceleration is always a straight line. The steeper the gradient the greater the acceleration. Distance travelled=area under speed-time graph. Distance=average speedxtime, so you can find the distance travelled by working out the area under a speed-time graph. Non-uniform acceleration is a curve on a V-T graph. If the acceleration is changing, the gradient of the velocity-time graph will also be changing, so you wont get a straight line. Increasing acceleration is shown by an increasing gradient, and decreasing acceleration is shown by a decreasing gradient. You can draw displacement-time and velocity-time graphs using ICT. Instead of using traditional methods to work out distance and time like a stopwatch and ruler, you can use ICT. For motion experiments a fairly standard piece of equipment is an ultrasound-position detector. This is a type of data-logger that automatically records the distance of an object from the sensor several times a second. If you attach one of these detectors to a computer with graph-drawing software, you can get real-time displacement-time and velocity-time graphs. The main advantages of doing this are: The data is more accurate, automatic systems have a much higher sampling rate than humans, and you can see the data displayed in real time. 

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