How does all of Newton's laws of motion apply to a bike?

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Newtons first law of motion is sometimes called the law of inertia.  Inertia is an objects tendency to resist changes in motion, so that could be interpreted in the example of the bicycle this way:  The bicycle, before it moves, has inertia.  It wants to remain still.  But, if the rider gets on the bike and exerts an outside force, such as pushing down on the bikes pedals, it will cause the bike to change what it is doing and move forward.

Newtons second law of motion is called the force law, and has a formula to calculate the amount of force an object like a bicycle can generate.  The formula is F=ma, where the F is the force generated, the m is the mass of the object, and the a is the acceleration of the object.  The bike has a constant mass, but if you change the amount of the acceleration the bike is experiencing (going slow versus going fast) you have two distinctly different amounts of force generated.

Finally, Newtons third law of motion is the action/reaction law, and basically says for every action, there is an equal and opposite reaction.  If you push down on the pedals of the bike, the bike goes forward.  If you sit on the bike, the bike pushes back against the weight of your body.  Equal and opposite reaction forces to the initial force given.

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