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Further Maths

  • A-Level Further Maths

Momentum, Impulse, Work, Energy & Power

As you know, momentum, p, is defined as the product of an object's mass and velocity:

p = mv

The Impulse-Momentum Principle

When an object collides with something else, it experiences a change in momentum. This is equal to the impulse experienced by that object in the collision:

I = Δmv

The impulse can also be calculated as the product of the force of the collision, F, and its duration, t:

I = Δmv = Ft
  • Therefore, the units of impulse are Newton-seconds, Ns.

Often, the force will have to be calculated using Newton's second law (F=ma), or SUVAT equations need to be used to calculate the duration or change in momentum.



The Principle of Conservation of Momentum

According to Newton's third law, for every reaction there is an equal but opposite reaction. This means that when two particles collide, both experience the same magnitude of impulse, but in opposite directions. Since impulse equals the change in momentum, the two particles experience equal and opposite changes in momentum, Δp and -Δp. Overall, these two changes in momentum cancel out:

the total momentum before impact equals the total momentum after the impact

Mathematically, this can be expressed as:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂


Work & Energy

Work done is defined as the the amount of energy transferred from one form to another when a force causes movement. This is known as the work-energy principle.

W = Fx Work done = force x distance moved.
  • The force and motion must be in the same direction

Because of this, you will often need to use trigonometry to find the correct components of forces:

The units of energy are joules, J - so when the force is given in Newtons, N, and the distance in metres, m, the work done has units joules, J.



Work Done Against Gravity & Gravitational Potential Energy

Work done against gravity is given by the equation:

W = mgh

This can easily be derived from the work-done equation above:

  • W = max: substitute F = ma into W = Fx

  • W = mah: for work done against gravity, we tend to refer to the distance, x, as height, h

  • W = mgh: acceleration due to gravity is g, 9.8 m/s²

According to the Work-Energy Principle, work done is equal to energy transferred. Therefore, gravitational potential (G.P.E.) is also defined with this equation:

G.P.E = mgh

Kinetic Energy

Kinetic Energy, KE, is given by the equation:

KE = ½mv²

This can also be derived from the work-done equation:

  • a = (v²-u²) / 2s: rearrange v² = u ²+ 2as to make a the subject

  • F = m(v²-u²) / 2s: substitute this into F = ma

  • Fs = ½m(v²-u²): multiply both sides by displacement, s

  • W = ½m(v²-u²): Fs = Fx = Work done

  • ΔKE = ½m(v²-u²): Work done = the change in energy

  • KE = ½mv²



Power

Power is defined as the rate at which work is done:

P = Fv

The units of power are Watts, W. If the force is given in Newtons and the velocity in m/s, the units of power are watts, 1 joule per second.



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