Force and Momentum

Momentum Facts

• The unit for momentum is kgms^-1
• Momentum is a vector, it always has a direction associated to it
• Momentum (p) = mv
• F = ma = m(v/t) = (mv)/t

Units and Usage

• P =MV gives standard unit for momentum
• Using the force formula showing force is change in momentum per unit time and rearranging gives the more useful unit Ns (Newton's second law)

Impulse

• The impulse of a force on an object is equal to the change of momentum of an object. Remember this could be due to mass and/or velocity changing.
• F = (mv)/t = Ft = mv

Force-Time Graphs

• In more complex systems where the force is not linear with time then a graph can be drawn to show how force changes over time
• The are under a force-time graph is therefore F*t (change in momentum or impulse of the force)

Changing Velocity

• F = (mv)/t = (m(v-u))/t = (mv-mu)/t
• Force is equal to change of momentum over time
• Any of the above formulae can be used depending on the data given

Perpendicular Rebounds

• If an object rebounds off a wall then the end velocity (v) is equal in magnitude but opposite in direction to initital velocity (u) (v = -u)
• The change in momentum is therefore: final momentum - inital momentum
• mv - mu = -2mu -> F = (-2mu)/t

Rebound Impacts

• All impact rebounds can be simplified as if the impact occured at 90° to the surface as there will be no change of momentum parallel to the surface.
• The component of the velocity perpendicular to the surface will need to be resolved
• F = (-2mu Cos(θ))/t

Changing Speed

• F = (M(v-u))/t
• Sometimes the speed of the object will change on rebound
• You must calculate the difference in velocities taking care of signs

Newton's third law of motion

• When two objects interact, they exert equal and opposite forces of each other (F1 = -F2)
• E.g. you sit on a stool and your weight pulls you down; the stool exerts a force on you equal tol your weight pushing you up.

Principle of Conservation of Momentum

• For a system of interacting bodies, the total momentum remains constant, provided no external resultant force acts on the system. (P1 = P2)

Objects moving in opposite direction

• Remember that momentum is a vector
• If objects are moving in opposite directions then a positive direction needs to be specified and one object needs a negative momentum
• If the two objects have identical momentums before the collision then they will both come to rest on impact

Elastic Collisions

• An elastic collision is one where there is no loss of kinetic energy
• Perfectly elastic objects must not yield at all or energy would be used up (and hence kinetic energy would be lost)
• Collisions between sub-atomic particles are an example of an elastic collision

Partially Inelastic Collision

• A partially inelastic collision is where the colliding objects move apart, and have less kinetic energy after the collision than before (e.g. dropping a tennis ball)

Totally Inelastic Collision

• A tottaly inelastic collision is one where the colliding object stick together (e.g. railway carriages)

Useful Formulae

• Epot = mgh (potential energy)
• Ekin = 1/2 mv^2 (kinetic energy)
• P = mv (momentum)
• g = 9.81Nkg^-1

Rules for collisions

• Kinetic energy is only conserved in elastic collisions
• Potential energy is converted to kinetic energy as an object falls; conversely as they rebound into the air
• Momentum is always conserved in all collisions

• Total inital momentum = 0
• Total momentum immediately after explosion: momentum of A + momentum of B = MaVa + MbVb
• Using principle of conservation of momentum: MaVa + MbVb = 0 => MbVb = -MaVa
• Minus sign means they move apart in opposite directions