Motion Newton’s Laws 1. A body in motion will remain in motion or a body at rest will remain at rest unless acted on by an external force. 2. An external force on a body of mass m causes an acceleration of that body, i.e. F = ma 3. For every action, there is an equal and opposite reaction. Derivation of Laws of Motion F = ma a = F/m Where a is the acceleration. Acceleration is defined as the derivative with respect to time of the velocity a = dv/dt = Δv/Δt Where both a and v are vectors. dv = a dt ∫dv = ∫adt If a = constant v = vo + at Where vo is the velocity at t=0. v = vo + at The definition of velocity is v = Δ x/Δ t = dx/dt dx/dt = v dx = v dt ∫dx = ∫v dt x = xo + vot + (1/2) at2 v = dx/dt v/a = (dx/dt) / (dv/dt) v/a = dx/dv vdv = adx ∫vdv = ∫adx v2 = (vo)2 + 2a(x – xo) One Dimensional Motion • Josh Avery climbs to the top of a 200 ft tall tree carrying two metal spheres – the first weighs 1 pound and the second weighs 10 pounds. He drops the lighter sphere and it falls to the ground. • What is the speed of the sphere when it impacts the ground? • How long did it take for the sphere to hit the ground? • Josh now drops the second and heavier sphere. • How fast is it going when it impacts the ground? • Is this faster than the lighter sphere? • How long does it take to hit the ground? • Is this less time than the lighter sphere? • Explain your answers. • A pitcher for the Boston Red Sox visits the campus at Dartmouth. He is asked to throw a baseball straight up as fast as he can. • The ball is clocked leaving the pitchers hand at 60 mph and at a height of 8 ft. • What is the force on the ball as it travels upwards? • What is the acceleration of the ball as it travels upwards? • At what height will the ball stop going up and begin to fall downward? • How fast will the ball be traveling when it changes direction? • How long did it take for the ball to reverse direction? • How fast will the ball be going when it hits the ground?