26 Mechanics 3

A knowledge of the topics and associated formulae from Modules Mechanics 1, Core 1 and Core 2 is required. A knowledge of the trigonometric identity $\space \sec^2x = 1 + \tan^2 \space x$ is also required. Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

Momentum and Collision
 $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ $m_1{\bf \mbox{u}}_1 + m_2{\bf \mbox{u}}_2 = m_1{\bf \mbox{v}}_1 + m_2{\bf \mbox{v}}_2$ $v = eu$ $v_1 - v_2 = -e(u_1 - u_2)$
 Relative velocity. Use of relative velocity and initial conditions to find relative displacement. Velocities may be expressed in the form $\space \mbox{a}\space {\bf \mbox{i}} + \mbox{b}\space {\bf \mbox{j}} + \mbox{c}\space {\bf \mbox{k}}\space$or as column vectors. Interception and closest approach. Use of calculus or completing the square. Geometric approaches may be required.
 Finding dimensions of quantities. Prediction of formulae. Checks on working, using dimensional consistency. Finding the dimensions of quantities in terms of M, L and T. Using this method to predict the indices in proposed formulae, for example, for the period of a simple pendulum. Use dimensional analysis to find units, and as a check on working.
 Momentum. Impulse as change of momentum. Knowledge and use of the equation  $I = mv - mu$. Impulse as Force $\times$ Time. Impulse as$\int F\space \mbox{d}t$ $I = Ft$  Applied to explosions as well as collisions. Conservation of momentum. Newton’s Experimental Law. Coefficient of restitution. $m_1u_1 + m_2u_2 = m_1v_1+m_2v_2$ $v=eu$ $v_1 - v_2 = -e(u_1-u_2)$
 Momentum as a vector. Impulse as a vector. ${\bf \mbox{I}} = m {\bf \mbox{v}} - m {\bf \mbox{u}} \space$ and $\space {\bf \mbox{ I }} = {\bf \mbox{F}}t \space$ will be required. Conservation of momentum in two dimensions. $m_1 {\bf \mbox{u}_1} + m_2 {\bf \mbox{u}_2} = m_1{\bf \mbox{v}_1} + m_2{\bf \mbox{v}_2}$ Coefficient of restitution and Newton’s experimental law. Impacts with a fixed surface. The impact may be at any angle to the surface. Candidates may be asked to find the impulse on the body. Questions that require the use of trigonometric identities will not be set. Oblique Collisions Collisions between two smooth spheres. Candidates will be expected to consider components of velocities parallel and perpendicular to the line of centres.
 Elimination of time from equations to derive the equation of the trajectory of a projectile Candidates will not be required to know the formula $\displaystyle y=x \tan \alpha-\frac{gx^2}{2v^2}(1+\tan^2 \alpha)$, but should be able to derive it when needed. The identity  $1+ \tan^2 \theta = \sec^2 \theta \space$ will be required.
 Projectiles launched onto inclined planes. Problems will be set on projectiles that are launched and land on an inclined plane. Candidates may approach these problems by resolving the acceleration parallel and perpendicular to the plane. Questions may be set which require the use of trigonometric identities, but any identities which are needed, apart from $\space \tan^2 \space x + 1 = \sec^2 \space x$ and those in Core 2, will be given in the examination paper. Candidates will be expected to find the maximum range for a given slope and speed of projection. Candidates may be expected to determine whether a projectile lands at a higher or lower point on the plane after a bounce.