Physics & Math Formula Sheet This is a reference guide for the key formulas for math & physics calculations used in game development Speed Speed is a scalar quantity that refers to "how fast an object is moving." Speed can be thought of as the rate at which an object covers distance. A fast-moving object has a high speed and covers a relatively large distance in a short amount of time. Contrast this to a slow-moving object that has a low speed; it covers a relatively small amount of distance in the same amount of time. An object with no movement at all has a zero speed. The math formula for speed is distance divided by time. Or simply put, the average speed or how fast something is moving is the distance it has traveled divided by the time it takes to travel. Velocity Velocity is similar to speed, but it has direction. When evaluating the velocity of an object, one must keep track of direction. It would not be enough to say that an object has a velocity of 55 mi/hr. One must include direction information in order to fully describe the velocity of the object. For instance, you must describe an object's velocity as being 55 mi/hr,east. This is one of the essential differences between speed and velocity. Speed is a scalar quantity and does not keep track of direction; velocity is a vector quantity and is direction aware. For example a swimmer may swim at the speed of 1 meter per second, but he or she swims at the velocity of 1 meter per second east. By specifying that he or she swims in the eastwardly direction, you are defining the velocity he or she swims at. So when you measure velocity, remember to report both speed and direction, the two components of velocity. The formula of velocity used in physics is the same as the formula used for measuring average speed per direction. Let's examine how the velocity formula works. In the diagram above, the three velocities represented are A, B, and C. A is the speed measured in the eastward direction, B is the speed measured in the southeast direction and C is the speed measured in the northeast direction. For example, velocity A is 10 m/s E, V is 20 m/s SE, and C is 15 m/s NE. Just remember to specify direction when reporting the velocity. Acceleration Acceleration is an expression of the change in the velocity of an object. Since the velocity has two components, speed and direction, the change can be in speed, direction, or both speed and direction. Since acceleration is defined as the change in velocity, acceleration is a vector quantity (as opposed to a scalar quantity) meaning acceleration is directional. Suppose you have a car that goes from 0 miles per hour to 60 miles per hour in 5 seconds. In standard unit, the speed 60 miles per hour is approximately 26.8 meters per second. Suppose that the acceleration is constant going from 0 m/s to 26.8 m/s in one direction. Then the acceleration is the change in speed or 5.36 m/s2. Notice that the unit for acceleration is meters per second squared because the acceleration is the rate of change in velocity. That's it. This is the easy way to calculate acceleration - just measure the change in velocity and divide that by the time taken to change the velocity. Newton's Laws of Motion Newton's Laws of Motions were invented by the famous English mathematician, physicist and astronomer Sir Isaac Newton. Among many other great discoveries, Sir Isaac Newton formulated three laws of motion which are still widely used today. Newton's Laws of Motion revolutionized science in the 17th Century. ● The first law of motion states that an object is at rest unless acted on by a force or an object moving at a constant speed will continue to move at that speed unless acted on by a force. ● The second law of motion introduces the formula Force equals mass times acceleration. ● The third law follows the law of conservation of momentum used widely in mathematics and states that each action has an equal and opposite reaction. Newton's First Law of Motion Newton's First Law is sometimes known as the Law of Inertia. The first part of this law states that unless acted on by an outside force, an object at rest stays at rest. The second part this law of motion states that unless acted on by an outside force, an object moving with uniform velocity continues to move at that same velocity. This isn’t to say that if you kick a ball, it will continue to move at the same speed forever until it hits a wall or other obstacle because it is acted on by many other smaller forces such as air resistance, friction, etc. Newton's Second Law Newton’s Second Law of Motion states that the acceleration of an object is dependent upon two variables the net force acting upon the object and the mass of the object. The formula is: Force = Mass x Acceleration The second law shows that if you exert the same force on two objects of different mass, you will get different accelerations (changes in motion). The acceleration on the smaller mass will be greater. The effect of a 10 newton force on a baseball would be much greater than that same force acting on a truck. The difference in effect (acceleration) is entirely due to the difference in their masses. Newton's Third Law Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. Forces are found in pairs. Think about the time you sit in a chair. Your body exerts a force downward and that chair needs to exert an equal force upward or the chair will collapse. It's an issue of symmetry. Acting forces encounter other forces in the opposite direction. Another example is shooting a cannonball. When the cannonball is fired through the air, the cannon is also pushed backward. The force pushing the ball out was equal to the force pushing the cannon back, but the effect on the cannon is less noticeable because it has a much larger mass. That example is similar to the kick when a gun fires a bullet forward.