Motion may be described and analysed by the use of graphs and equations
- Distance and displacement
- Speed and velocity
- Graphs describing motion
- Equations of motion for uniform acceleration
- Projectile motion
- Fluid resistance and terminal speed
Applications and Skills:
- Determining instantaneous and average values for velocity, speed and acceleration
- Solving problems using equations of motion for uniform acceleration
- Sketching and interpreting motion graphs
- Determining the acceleration of free-fall experimentally
- Analysing projectile motion, including the resolution of vertical and horizontal components of acceleration, velocity and displacement
- Qualitatively describing the effect of fluid resistance on falling objects or projectiles, including reaching terminal speed
MOTION IN A STRAIGHT LINE
DISTANCE AND DISPLACEMENT
- Displacement may be positive, negative or zero but distance is always positive.
- Displacement is not affected by the shift of the coordinate axes.
- Displacement of an object is independent of the path followed by the object but distance depends upon path.
- Displacement and distance both have same unit as that of length i.e. metre.
- For a moving body distance always increases with time
- For a body undergoing one dimensional motion, in the same direction distance = | displacement |. For all other motion distance > | displacement |.
- | Average velocity | can be zero but average speed cannot be zero for a moving object.
- | Instantaneous velocity | = Instantaneous speed.
- A particle may have constant speed but variable velocity. It happens when particle travels in curvilinear path.
- If the body covers first half distance with speed v1 and next half with speed v2 then
- If a body covers first one-third distance at a speed v1, next one-third at speed v2 and last one-third at speed v3, then
- If a body travels with uniform speed v1 for time t1 and with uniform speed v2 for time t2, then
EQUATIONS FOR UNIFORMLY ACCELERATED MOTION
- v = u + at
- v2 – u2 = 2as
v = final velocity
s = distance travelled in time t
- Distance travelled in nth second
sn = ;
VERTICAL MOTION UNDER GRAVITY
- For a body thrown downward with initial velocity u from a height h, the equations of motion are
- If initial velocity is zero, then the equations are
- When a body is thrown upwards with initial velocity u, the equations of motion are
UNIFORMLY ACCELERATED MOTION : A DISCUSSION
- The direction of average acceleration vector is the direction of the change in velocity vector
- There is no definite relationship between velocity vector and acceleration vector.
- For a body starting from rest and moving with uniform acceleration, the ratio of distances covered in t1 sec.,
t2 sec, t3 sec, etc. are in the ratio t12 : t22 : t32 etc.
- A body moving with a velocity v is stopped by application of brakes after covering a distance s. If the same body moves with a velocity nv, it stops after covering a distance n2s by the application of same retardation.
- An object moving under the influence of earth’s gravity in which air resistance and small changes in g are neglected is called a freely falling body.
- In the absence of air resistance, the velocity of projection is equal to the velocity with which the body strikes the ground.
- Distance travelled by a freely falling body in 1st second is always half of the numerical value of g or 4.9 m, irrespective of height h.
- For a freely falling body with initial velocity zero
- Velocity ∝ time (v = gt)
- Velocity (v2 = 2gs)
- Distance fallen α (time)2 , where g is the acceleration due to gravity.
- If maximum height attained by a body projected vertically upwards is equal to the magnitude of velocity of projection, then velocity of projection is 2g ms–1 and time of flight is 4 sec.
- If maximum height attained by a body projected upward is equal to magnitude of acceleration due to gravity i.e., ‘g’, the time of ascent is √2 sec. and velocity of projection is g√2.
- Ratio of maximum heights reached by different bodies projected with velocities u1, u2, u3 etc. are in the ratio of etc. and ratio of times of ascent are in ratio of u1 : u2 : u3 etc.
- During free fall velocity increases by equal amount every descend and distance covered during 1st, 2nd, 3rd seconds of fall, are 4.9m, 14.7m, 24.5m.
- If a body is projected horizontally from top of a tower, the time taken by it to reach the ground does not depend on the velocity of projection, but depends on the height of tower and is equal to .
- If velocity v of a body changes its direction by θ without change in magnitude then the change in velocity will be .
- From the top of a tower a body is projected upward with a certain speed, 2nd body is thrown downward with same speed and 3rd is let to fall freely from same point then
t2 = time taken by the body thrown downward and
t3 = time taken by the body falling freely.
- If a body falls freely from a height h on a sandy surface and it buries into sand upto a depth of x, then the retardation produced by sand is given by .
- In case of air resistance, the time of ascent is less than time of descent of a body projected vertically upward i.e. ta < td.
- When atmosphere is effective, then buoyancy force always acts in upward direction whether body is moving in upward or downward direction and it depends on volume of the body. The viscous drag force acts against the motion.
- If bodies have same volume but different densities, the buoyant force remains the same.
- When an aeroplane flying horizontally drops a bomb.
- An ascending helicopter dropping a food packet.
- A stone dropped from a moving train etc.
VARIOUS GRAPHS RELATED TO MOTION
- For a stationary body (v = 0) the time-displacement graph is a straight line parallel to time axis.
- When the velocity of a body is constant then time-displacement graph will be an oblique straight line. Greater the slope of the straight line, higher will be the velocity.
- If the velocity of a body is not constant then the time-displacement curve is a zig-zag curve.
- For an accelerated motion the slope of time-displacement curve increases with time while for decelerated motion it decreases with time.
- When the particle returns towards the point of reference then the time-displacement line makes an angle θ > 90° with the time axis.
- When the velocity of the particle is constant or acceleration is zero.
- When the particle is moving with a constant acceleration and its initial velocity is zero.
- When the particle is moving with constant retardation.
- When the particle moves with non-uniform acceleration and its initial velocity is zero.
- When the acceleration decreases and increases.
- The total area enclosed by the time – velocity curve represents the distance travelled by a body.
- When the acceleration of the particle is zero.
- When acceleration is constant
- When acceleration is increasing and is positive.
- When acceleration is decreasing and is negative
- When initial acceleration is zero and rate of change of acceleration is non-uniform
- The change in velocity of the particle = area enclosed by the time-acceleration curve.
RELATIVE VELOCITY (IN ONE DIMENSION)
⇒ vAB = vA + vB
MOTION IN A PLANE
SCALARS AND VECTORS
- it must have magnitude.
- it must have direction.
- it must satisfy parallelogram law of vector addition.
TYPES OF VECTORS
Properties of Null or Zero Vector :
- The sum of a finite vector and the zero vector is equal to the finite vector
- The multiplication of a zero vector by a finite number n is equal to the zero vector
- The multiplication of a finite by a zero is equal to zero vector
LAWS OF VECTOR ALGEBRA
ADDITION OF VECTORS
TRIANGLE LAW OF VECTOR ADDITION
PARALLELOGRAM LAW OF VECTOR ADDITION
POLYGON LAW OF VECTOR ADDITION
- Resultant of two unequal vectors cannot be zero.
- Resultant of three coplanar vectors may or may not be zero.
- Minimum no. of coplanar vectors for zero resultant is 2 (for equal magnitude) and 3 (for unequal magnitude).
- Resultant of three non coplanar vectors cannot be zero. Minimum number of non coplanar vectors whose sum can be zero is four.
- Polygon law should be used only for diagram purpose for calculation of resultant vector (For addition of more than 2 vectors) we use components of vector.
KEEP IN MEMORY
- A vector can be divided or multiplied by a scalar.
- Vectors of the same kind can only be added or subtracted. It is not possible to add or subtract the vectors of different kind. This rule is also valid for scalars.
- Vectors of same as well as different kinds can be multiplied.
- A vector can have any number of components. But it can have only three rectangular components in space and two rectangular components in a plane. Rectangular components are mutually perpendicular.
- The minimum number of unequal non-coplanar whose vector sum is zero is 4.
SUBTRACTION OF VECTORS
RESOLUTION OF A VECTOR
RECTANGULAR COMPONENTS OF A VECTOR IN PLANE
RECTANGULAR COMPONENTS OF A VECTOR IN 3D
Do not resolve the vector at its head.
PRODUCT OF TWO VECTORS
SCALAR OR DOT PRODUCT
PROPERTIES OF SCALAR OR DOT PRODUCT
- .= A (B cosθ) = B (A cosθ)
- Dot product of two vectors is commutative.
- Dot product is distributive.
- = (Ax Bx + Ay By + Az Bz)
VECTOR OR CROSS PRODUCT
PROPERTIES OF VECTOR OR CROSS PRODUCT
- The cross product of two vectors represents the area of the parallelogram formed by them.
- A unit vector which is perpendicular to A as well as B is
Division by a vector is not defined. Because, it is not possible to divide by a direction.
CONDITION OF ZERO RESULTANT VECTOR
MOTION IN A PLANE OR MOTION IN TWO DIMENSIONS
RELATIVE VELOCITY IN TWO DIMENSIONS
- Relative velocity of A w.r.t B
- Relative velocity of B w.r.t. A
- A uniform velocity in the horizontal direction, which does not change (if there is no air resistance)
- A uniformly changing velocity in the vertical direction due to gravity.
TYPES OF PROJECTILE
- Oblique projectile : In this, the body is given an initial velocity making an angle with the horizontal and it moves under the influence of gravity along a parabolic path.
- Horizontal projectile : In this, the body is given an initial velocity directed along the horizontal and then it moves under the influence of gravity along a parabolic path.
EQUATION OF TRAJECTORY
- The horizontal range of the projectile is same at two angles of projection for θ and (90° – θ).
- The height attained by the projectile above the ground is the largest when the angle of projection with the horizontal is 90° (vertically upward projection). In such a case time of flight is largest but the range is the smallest (zero).
- If the velocity of projection is doubled. The maximum height attained and the range become 4 times, but the time of flight is doubled.
- When the horizontal range of the projectile is maximum, (θ = 45°), then the maximum height attained is ¼th of the range.
- For a projectile fired from the ground, the maximum height is attained after covering a horizontal distance equal to half of the range.
PROJECTILE ON AN INCLINED PLANE
- Equation of trajectory of an oblique projectile in terms of range (R) is
- There are two unique times at which the projectile is at the same height h(< H) and the sum of these two times.
(t1 + t2) is and product (t1t2) is .
UNIFORM AND NON-UNIFORM CIRCULAR MOTION
UNIFORM CIRCULAR MOTION
NON-UNIFORM CIRCULAR MOTION
- Angular displacement behaves like vector, when its magnitude is very small. It follows laws of vector addition.
- Angular velocity and angular acceleration are axial vectors.
- Centripetal acceleration always directed towards the centre of the circular path and is always perpendicular to the instantaneous velocity of the particle.
- Circular motion is uniform if aT = rα = 0, that is angular velocity remains constant and radial acceleration is constant.
- When aT or α is present, angular velocity varies with time and net acceleration is
- If aT = 0 or α = 0, no work is done in circular motion.