Regents Physics momentum, energy and gravitation: a complete skills guide to impulse, conservation of momentum, work, energy conservation, circular motion and gravitation

Why the conservation laws are the most powerful tools in mechanics

Once you can track momentum and energy, many problems that look hard become a single equation: set a conserved quantity equal before and after. This module brings together the two great conservation laws, momentum and energy, along with the forces (circular and gravitational) that govern motion on the scale of vehicles and planets. The guide ties together the dot-point pages, each with its own practice: momentum and impulse, conservation of momentum, work and power, energy and its conservation, uniform circular motion and universal gravitation.

Momentum and impulse

Momentum is $p = mv$, a vector in kg m/s. Impulse is $F\Delta t$ in newton seconds, and the impulse-momentum relationship $F\Delta t = \Delta p$ (both printed on the Reference Tables) says the impulse equals the change in momentum. Rearranged, $F = \dfrac{\Delta p}{\Delta t}$: for a fixed momentum change, a longer contact time means a smaller force. This is why airbags and crumple zones save lives. Track signs carefully, because a rebounding object reverses its velocity, giving a larger momentum change than one that merely stops.

Conservation of momentum

In an isolated system the total momentum is conserved: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$. It follows from Newton's third law, since the equal and opposite collision forces act for the same time, giving equal and opposite impulses. Use it for collisions (set total momentum before equal to after; if the objects stick, the right side is $(m_1 + m_2)v_f$) and for explosions or recoils (the total is often zero, so fragments carry equal and opposite momenta). This relationship is not on the Reference Tables, so recall it.

Work, energy and the conservation of energy

Work is $W = Fd$ (force along the displacement), in joules, and equals the energy transferred ($W = \Delta E_T$). A perpendicular force does no work. Power is $P = \dfrac{W}{t} = Fv$, in watts. The forms of mechanical energy are kinetic ($KE = \tfrac{1}{2}mv^2$), gravitational potential ($PE = mgh$) and elastic ($PE_s = \tfrac{1}{2}kx^2$), all printed on the tables. The conservation of energy lets you equate total mechanical energy at two points in a frictionless system, $KE_i + PE_i = KE_f + PE_f$. With friction, mechanical energy is converted to internal (thermal) energy, but the total energy is still conserved.

Uniform circular motion and gravitation

Uniform circular motion is accelerated even at constant speed, because the velocity direction changes. The centripetal acceleration is $a_c = \dfrac{v^2}{r}$ and the centripetal force is $F_c = \dfrac{mv^2}{r}$ (both on the tables), directed toward the center and supplied by a real force (tension, gravity, friction, normal force). There is no outward force. Universal gravitation is the inverse-square attraction $F_g = \dfrac{Gm_1 m_2}{r^2}$: doubling the distance quarters the force. Near a surface this is the weight $F_g = mg$, with $g = \dfrac{GM}{r^2}$. Gravity supplies the centripetal force that holds satellites in orbit.

Check your knowledge

A mix of recall, calculation and reasoning questions covering the module. Attempt them under timed conditions, then check against the solutions.

1. A $3.0$ kg object moves at $5.0$ m/s. Calculate its momentum. (2 marks)
2. Explain why a longer stopping time reduces the force in a collision. (2 marks)
3. State the law of conservation of momentum. (2 marks)
4. A $2.0$ kg cart at $4.0$ m/s strikes a $2.0$ kg cart at rest and they stick. Calculate the common velocity. (2 marks)
5. A $60.$ N force moves an object $4.0$ m in the direction of the force. Calculate the work done. (2 marks)
6. A $2.0$ kg object moves at $3.0$ m/s. Calculate its kinetic energy. (2 marks)
7. State what happens to mechanical energy lost to friction. (1 mark)
8. State why an object in uniform circular motion is accelerating. (2 marks)
9. A $1.0$ kg ball moves in a circle of radius $2.0$ m at $4.0$ m/s. Calculate the centripetal force. (2 marks)
10. State how the gravitational force changes if the distance between two masses is doubled. (1 mark)