AH Mechanics Checklist (Unit 1) AH Mechanics Checklist (Unit 1) Rectilinear Motion

Transcription

1 Rectilinear Motion No. kill Done 1 Know that rectilinear motion means motion in 1D (i.e. along a straight line) Know that a body is a physical object 3 Know that a particle is an idealised body that has no extent, i.e., assumed to be a mathematical point 4 Know that uniform means unchanging or constant (wrt time) 5 Know that a scalar quantity is one that has magnitude only 6 Know that a vector quantity is one that has magnitude and direction 7 Know that a reference frame is a system of coordinates used to describe the motion of a body 8 Know that position is the location of an object relative to a reference frame 9 Know that the I unit of time is the base unit the second (s) 10 Know that displacement from the origin is a function of time, where u is a unit vector (normally taken to be i, j or k) in the direction of motion: s (t) s (t) u x (t) u x (t) 11 Know that displacement is a vector quantity 1 Know that distance is the magnitude of displacement: s (t) s x (t) 13 Know that distance is a scalar quantity 14 Know that the I unit of displacement or distance is the base unit the metre (m) 15 Calculate distance in 1D given displacement 16 Know that, in 1D, constant displacement is equivalent to constant distance 17 Know that velocity is the first derivative of displacement: ds v (t) & s (t) s&(t) u dt 18 Know that velocity is a vector quantity 19 Calculate velocity in 1D given displacement 0 Know that displacement is the integral of velocity: s (t) v (t) dt v (t) dt u M atel (eptember 01) 1 t. Machar Academy

2 1 Calculate displacement in 1D given velocity Know that speed is the magnitude of velocity: v (t) v s&(t) x& (t) 3 Know that speed is a scalar quantity 4 Know that the I unit of velocity or speed is the derived unit m s 1 5 Calculate speed in 1D given velocity 6 Know that, in 1D, constant velocity is equivalent to constant speed 7 Know that an object with zero velocity has constant displacement 8 Know that at an instant means at a specific value of t 9 Know that initially means at the start of motion, normally t 0 30 Know that at rest (when t a) means zero velocity (equivalently, zero speed) at t a 31 Know that the acceleration vector is the first derivative of velocity (equivalently, the second derivative of displacement): dv d s a (t) v &(t) && s (t) dt dt 3 Know that the acceleration vector is (obviously) a vector 33 Calculate acceleration in 1D given velocity or displacement 34 Know that velocity is the integral of acceleration: v (t) a (t) dt a (t) dt u 35 Calculate velocity in 1D given acceleration 36 Know that the magnitude of acceleration is the magnitude of the acceleration vector: a (t) a v&(t) s&&(t) x&&(t) 37 Know that the magnitude of acceleration is a scalar 38 Know that the I unit of acceleration is the derived unit m s 39 Know that the acceleration vector and magnitude of acceleration are often both referred to as acceleration (the context is usually self-evident) 40 Calculate acceleration in 1D given the acceleration vector 41 Know that, in 1D, a constant acceleration vector is equivalent to constant magnitude of acceleration 4 Know that an object with zero acceleration moves with constant velocity 43 Know that every body creates a gravitational field due to its mass 44 Know that in the gravitational field of a given body, the acceleration M atel (eptember 01) t. Machar Academy

3 due to gravity, g, describes how fast any object in that field will accelerate in vacuum (i.e., ignoring all resistive forces) 45 Know that for a given gravitating body, g generally decreases the further away from the body 46 Know that the acceleration due to gravity of a given body is assumed to be constant close to the body s surface 47 Know that near the Earth s surface, the acceleration due to gravity is approximated as 9 8 m s towards the Earth s centre, i.e., g g j 9 8 j 48 Know that the relationship between displacement, velocity and acceleration can be summarised as: s (t) d dx v (t) M atel (eptember 01) 3 t. Machar Academy d dx 49 Draw a displacement-time graph a (t) 50 Draw a distance-time graph from a displacement-time graph 51 Know that the gradient of a displacement-time graph at time t gives the velocity at t 5 Calculate velocity using a displacement-time graph 53 Know that the gradient of a distance-time graph at time t gives the speed at t 54 Calculate speed using a distance-time graph 55 Draw a velocity-time graph 56 Draw a speed-time graph from a velocity-time graph 57 Know that the gradient of a velocity-time graph at time t gives the acceleration vector at t 58 Calculate the acceleration vector using a velocity-time graph 59 Know that the gradient of a speed-time graph at time t gives the acceleration at t 60 Calculate acceleration using a speed-time graph 61 Know that the area under a velocity-time graph between t t1 and t t, where t 1 < t, gives the displacement covered between t 1 and t 6 Calculate displacement covered using a velocity-time graph 63 Know that the area under a speed-time graph between t t1 and t t, where t 1 < t, gives the distance covered between t 1 and t 64 Calculate distance travelled using a speed-time graph 65 Know that there are 4 Equations of Motion (EoM) that describe the motion of a body undergoing constant acceleration 66 Derive, using calculus, the 1 st EoM, which describes a body travelling

4 with constant acceleration a from an initial velocity u to a final velocity v over time t : v u + at 67 olve problems using the previous equation 68 Derive, using calculus, the nd EoM, which describes a body travelling a distance s with constant acceleration a over time t from an initial velocity u (where the initial value of s is 0) : s ut + at 69 olve problems using the previous equation 70 Derive, using the 1 st and nd EoM, the 3 rd EoM, which describes a body travelling a distance s with constant acceleration a from initial velocity u to final velocity v : 1 v u + as 71 olve problems using the previous equation 7 Derive, using the nd and 3 rd EoM, the 4 th EoM, which describes a body travelling a distance s with constant acceleration a over time t from initial velocity u to final velocity v : s 1 u + v t ( ) 73 olve problems using the previous equation 74 Know that the 1 st EoM is the one independent of s 75 Know that the nd EoM is the one independent of v 76 Know that the 3 rd EoM is the one independent of t 77 Know that the 4 th EoM is the one independent of a 78 olve problems involving non-constant acceleration M atel (eptember 01) 4 t. Machar Academy

5 Relative Motion in D and 3D No. kill Done 1 Know that motion in dimensions greater than 1 is best described through the use of vectors Know that the resultant of or more vectors is the vector sum of those vectors 3 Find the resultant of or more vectors in D or 3D 4 Resolve a given vector in D into vectors that are perpendicular to each other, especially into its horizontal and vertical components 5 Know that relative motion refers to motion between different bodies 6 Know that a rectangular coordinate system is a coordinate system consisting of (or 3 mutually) perpendicular axes with unit vectors i and j (and k) 7 Know that, relative to an origin O, the position vector of a body relative to O is the vector r (t) ined as: r (t) x (t) i + y (t) j r (t) or x (t) i + y (t) j + z (t) k 8 Know that distance is the magnitude of displacement and given by: r (t) r x + y or r (t) r x + y + z 9 Calculate distance in D or 3D given displacement 10 Know that, relative to an origin O, the velocity vector of a body relative to O is the vector v (t) ined as: v (t) dr dt r& (t) x& (t) i + y& (t) j or M atel (eptember 01) 5 t. Machar Academy

6 dr v (t) r& (t) x& (t) i + y& (t) j + z& (t) k dt 11 Calculate velocity in D or 3D given displacement 1 Know that displacement is the integral of velocity: r (t) v (t) dt x & dt i + or y & dt j r (t) v (t) dt x & dt i + y & dt j + z & dt k 13 Calculate displacement in D or 3D given velocity 14 Know that speed is the magnitude of velocity and given by: v (t) v x& + & y or v (t) v x& + & y + z& 15 Calculate speed in D or 3D given velocity 16 Know that, relative to an origin O, the acceleration vector of a body relative to O is the vector a (t) ined as: a (t) dv dt v& (t) r&& (t) x&&(t) i + y&&(t) j or dv a (t) v& (t) r&& (t) x&&(t) i + y&&(t) j + z&&(t) k dt 17 Calculate acceleration in D or 3D given velocity or displacement 18 Know that velocity is the integral of acceleration: v (t) a (t) dt x && dt i + or y && dt j M atel (eptember 01) 6 t. Machar Academy

7 v (t) a (t) dt x && dt i + y && dt j + z && dt k 19 Calculate velocity in D or 3D given acceleration 0 Know that acceleration is the magnitude of the acceleration vector and given by: a (t) a && x + && y or a (t) a && x + && y + && z 1 Calculate acceleration in D or 3D given the acceleration vector Given an origin O and two bodies and Q, the relative position vector (aka relative displacement vector) of Q with respect to is: uuur Q Q r r r Q 3 Calculate the relative position vector of objects given their individual position vectors in component form 4 Given an origin O and two bodies and Q, the relative velocity vector of Q with respect to is: Q v v v r& r& Q Q 5 Calculate the relative velocity vector of objects given their individual velocity vectors in component form 6 Calculate the relative velocity vector of bodies given the relative position vector in component form 7 Given an origin O and two bodies and Q, the relative acceleration vector of Q with respect to is: Q a a a v& v& Q Q r&& r&& Q 8 Calculate the relative acceleration vector of objects given their individual acceleration vectors in component form 9 Calculate the relative acceleration vector of bodies given the relative velocity vector or relative position vector in component form 30 Know that a nautical mile (aka international nautical mile), symbolised by M or NM, is a unit of length ined as 1 85 metres: M atel (eptember 01) 7 t. Machar Academy

8 1 NM 1 85 m 31 Know that a knot (aka international knot), symbolised by kn, is a unit of speed ined as 1 nautical mile per hour (which exactly equals 1 85 km h 1, approximately equals m s 1 and approximately equals mph): 1 kn 1 NM h 1 3 olve problems involving winds or currents using trigonometry, for example, given a wind blowing due North and a plane flying due East, calculate the bearing, resultant velocity and speed of the plane 33 Know that a collision course is a situation when or more objects will have the same position vector at some time 34 For a collision problem, given the initial relative position vector of objects and their respective constant velocity vectors, show that they are on a collision course 35 For a collision problem, given the initial relative position vector of objects and their respective constant velocity vectors, find the time when they collide 36 Know that a problem involving objects that have a closest (non-zero) distance is called a nearest approach problem (aka closest approach problem) 37 For a nearest approach problem, given the initial relative position vector of objects and their respective constant velocity vectors, find the time when they are closest 38 For a nearest approach problem, given the initial relative position vector of objects and their respective constant velocity vectors, find their closest distance M atel (eptember 01) 8 t. Machar Academy

9 rojectile Motion No. kill Done 1 Know that a projectile is any object that is thrown, dropped, launched etc. in a gravitational field Know that a projectile acted only upon by a gravitational field (i.e. no resistive forces) is called a free projectile 3 Know that, in this course, only free projectile motion in a vertical plane will be considered 4 Know that the velocity of projection of a projectile is the initial velocity with which the projectile is fired 5 Know that the angle of projection of a projectile is the angle the initial velocity vector makes with the positive x - axis 6 Given the velocity of projection and the angle of projection of a free projectile, obtain the horizontal and vertical velocity components 7 Know that the acceleration of a free projectile in a gravitational field with constant gravitational acceleration g is: r&& g j 8 Know that to solve the above equation means to obtain r 9 olve the previous equation for a free projectile fired from initial position c ( x, y ) with velocity of projection u ( r&) 0 0 and angle of projection θ to obtain: r ( x + ut cos θ 0 ) i + 1 ( y ut sin θ gt 0 ) + j 10 Know that, if the initial position is at the origin O of the coordinate system, the previous equation becomes: 1 r ( ut cos θ ) i + ( ut sin θ gt ) 11 Know that the trajectory of a projectile is its path 1 rove that the equation of the trajectory, y (x), of a free projectile fired from initial position c ( x, y ) with velocity of projection 0 0 u ( r&) and angle of projection θ is a parabola: j y g sec θ u x + tan θ + gx sec θ 0 x u + y x θ 0 tan Know that the previous trajectory equation reduces, M atel (eptember 01) 9 t. Machar Academy gx sec u θ

10 in the case of initial position at O, to: y g sec θ u x + ( tan θ ) x 14 Know that, for the initial position at O case, the horizontal displacement component is: x ut cos θ 15 olve problems using the previous equation 16 Know that, for the initial position at O case, the vertical displacement component is: y ut sin θ gt 17 olve problems using the previous equation 18 Know that the time of flight t of a projectile is the time the flight projectile takes to go from its initial to final positions 19 rove that the time of flight of a free projectile projected from the origin of the coordinate system with velocity of projection u and angle of projection θ back to vertical height 0 is: 1 t flight u sin θ g 0 Use the previous equation to solve problems 1 Know that the range of a projectile is the horizontal distance travelled from its initial to final positions rove that the range of a free projectile projected from the origin of the coordinate system with velocity of projection u and angle of projection θ back to vertical height 0 (aka horizontal range x ) is: range x ut cos θ range flight u sin θ g 3 olve problems using the previous equation 4 Use the trajectory equation to obtain x range 5 how that the maximum range is: 6 x max. rove that the angle needed to achieve x is π max. 4 radians 7 Know that the maximum height of a projectile is the M atel (eptember 01) 10 t. Machar Academy u g

11 greatest vertical distance the projectile reaches 8 rove, using calculus, that the time taken to reach maximum height t of a free projectile projected from the origin of the max. height coordinate system with velocity of projection u and angle of projection θ is: t max. height u sin g 9 rove the above equation using symmetry 30 olve problems using the previous equation 31 rove, using t, that the maximum height y of a free max. height max. projectile projected from the origin of the coordinate system with velocity of projection u and angle of projection θ is: θ y max. u sin θ g 3 rove the above equation using the trajectory equation and calculus 33 olve problems using the previous equation 34 olve projectile problems using a combination of the above equations M atel (eptember 01) 11 t. Machar Academy

12 Force, Newton s Laws of Motion and Friction No. kill Done 1 Know that mass characterises how much matter there is in an object Know that mass is a scalar quantity 3 Know that the I unit of mass is the base unit the kilogram (kg) 4 Know that momentum (aka linear momentum or translational momentum) of an object is the vector quantity ined as the product of its mass and velocity: p m v 5 Know that momentum is a vector quantity 6 Know that the I unit of momentum is the derived unit kg m s 1 7 olve problems using the previous equation 8 Know that a force is any influence that causes an object to change either its position or shape 9 Know that force is a vector quantity 10 Know that the I unit of force is the derived unit the Newton (N), equivalently kg m s 11 Know that weight is the gravitational force acting on a mass m, where the gravitational field strength in the vicinity of the object is g (not necessarily on Earth), and given by: W m g 1 olve problems using the previous equation 13 Resolve a force in D into perpendicular components (usually horizontally and vertically) 14 Know that a free-body diagram (aka force diagram) is a diagram showing all the forces acting on a body 15 Know that a free-body diagram does not show the term m a 16 Know that a free-body diagram does not show forces that the body acts on other objects 17 Find the resultant force (aka net force) acting on an object 18 Know that or more forces on an object are balanced if the resultant force on it is 0 19 Know that an object is in equilibrium if the forces on it are balanced 0 Know that the unbalanced force (on an object) is the resultant non-zero force (acting on the object) 1 Know that the natural motion of an object is its motion in a vacuum Know that a resistive force is one that opposes the natural motion of an object M atel (eptember 01) 1 t. Machar Academy

13 3 Give examples of resistive forces 4 Know that a smooth surface is one where there is no friction between it and any object placed on it 5 Know that a rough surface is one where there is friction between it and any object placed on it 6 Know that Newton s 1 st Law of Motion states that an object at rest or moving with uniform velocity will continue in that state unless acted upon by a non-zero net force 7 Know that an alternative formulation of Newton s 1 st Law is that if the net force on an object is zero, the object will move with constant velocity (which may be zero) 8 Know that Newton s nd Law of Motion states that the net force acting on an object is the rate of change of momentum wrt time: F d p dt 9 olve problems using the previous equation 30 Derive the more familiar form of Newton s nd Law from the above: F m a 31 olve problems using the previous equation 3 Know that Newton s 3 rd Law of Motion states that to every action, there is an equal and opposite reaction 33 Know that Newton s 3 rd Law is alternatively stated as, for any force F acting from body A to body B, there is an equal and opposite force F acting from body B to body A 34 Know that the forces referred to in Newton s 3 rd Law are termed an action-reaction pair 35 Know that an equilibrium problem is one that involves one or more bodies in equilibrium 36 Know that equilibrium problems are usually solved by resolving forces horizontally and vertically 37 olve equilibrium problems using Newton s 1 st and 3 rd Laws 38 Know that the normal force between surfaces is the force that is perpendicular to both surfaces and opposite to the component of the weight (which is perpendicular to the plane surface made by the contacting bodies) of one body on another 39 Know that, in the case of an object on a horizontal plane, the normal force is equal in magnitude to the object s weight and acts upwards (opposite to the weight) 40 Draw a free-body diagram for an object of mass m in a gravitational field of strength g placed on a horizontal plane M atel (eptember 01) 13 t. Machar Academy

14 41 Draw a free-body diagram for an object of mass m in a gravitational field of strength g projected with speed V up a smooth plane inclined at angle θ to the horizontal 4 Given an object of mass m in a gravitational field of strength g projected with speed V up a smooth plane inclined at angle θ to the horizontal, show that its acceleration a is: a g sin θ 43 how that, for the above situation, the distance s the mass travels up the plane before momentarily coming momentarily to rest is: s V g sin θ 44 olve problems using the previous equation 45 Know that friction is a resistive force that resists the relative motion of objects 46 Know that, generally, different frictional forces act between objects that have zero relative motion and non-zero relative motion 47 Know that static friction is the friction that exists when objects have no relative motion 48 Know that, for static friction, the frictional force that stops the objects from moving is called the static frictional force F 49 Know that the coefficient of static friction µ is ined as the ratio of the static frictional force magnitude to the normal force magnitude: F µ N 50 Know that µ is a dimensionless quantity and therefore has no units 51 Know that the (magnitude of the) static frictional force F between objects is given by the inequality: F µ N 5 olve problems using the previous inequation 53 Given µ and N, show that the maximum static frictional force ( F is: ) max. ( F ) max. M atel (eptember 01) 14 t. Machar Academy µ N 54 olve problems using the previous equation 55 Know that limiting friction is the frictional force required to just move the object and is therefore equal to ( F ) max.

15 56 Know that kinetic (aka dynamic or sliding) friction is the friction that exists when objects are in relative motion 57 Know that, for dynamic friction, the frictional force that exists when objects are in relative motion is called the dynamic frictional force F K 58 Know that the coefficient of dynamic friction µ is ined as the K ratio of the dynamic frictional force magnitude to the normal force magnitude: µ K N 59 Know that µ is a dimensionless quantity and therefore has no units K 60 Know that the (magnitude of the) kinetic frictional force F K between objects is given by the equation: F K F K µ N 61 olve problems using the previous equation 6 Know that, generally, µ is greater than µ K 63 Know that, when an object is on a rough plane that is gradually tilted until the block is on the point of moving down the plane, (i.e. in a state of limiting friction), the angle θ at which this point occurs is called the angle of friction (aka friction angle) 64 Draw a free-body diagram for the above situation 65 how that, in the above situation, θ and µ are linked by: K µ tan θ 66 olve problems using the previous equation 67 Draw a free-body diagram for an object of mass m in a gravitational field of strength g accelerating from rest down a rough plane inclined at angle θ to the horizontal reaching the bottom of the plane with speed V 68 Given an object of mass m in a gravitational field of strength g accelerating from rest down a rough plane inclined at angle θ to the horizontal reaching the bottom of the plane with speed V, where µ is the coefficient of dynamic K friction between the mass and the plane, show that its acceleration down the plane a is: a g ( sin θ µ cosθ K ) 69 how that, for the above situation, the distance L M atel (eptember 01) 15 t. Machar Academy

16 the mass travels down the plane is: L V ( K ) g sin θ µ cosθ 70 olve problems using the previous equation 71 Draw a free-body diagram for an object of mass m in a gravitational field of strength g projected with speed V up a rough plane inclined at angle θ to the horizontal 7 Given an object of mass m in a gravitational field of strength g projected with speed V up a rough plane inclined at angle θ to the horizontal, where µ is the coefficient of dynamic K friction between the mass and the plane, show that its acceleration up the plane a is: a g ( sin θ + µ cosθ K ) 73 how that, for the above situation, the distance L the mass travels up the plane before momentarily coming to rest is: L V ( + K ) g sin θ µ cosθ 74 olve problems using the previous equation M atel (eptember 01) 16 t. Machar Academy