Kinematics

Concept 1: Deriving equations which represent uniformly accelerated motion in a straight line using definitions of acceleration and velocity.

Step 1: Define acceleration

Acceleration is the rate of change of velocity. The direction of acceleration is that of velocity change. Thus from this definition,

Step 2: Look at v/t graph for uniformly accelerated motion

The area under the v/t graph gives the displacement.

Concept 2: Describing an object falling in a uniform gravitational field with air resistance

• When object starts to move, since speed is zero, there will be no drag force acting on the body. Since the only force is weight, the acceleration of the body is g.
• As speed increases, drag force also increases. Thus the net force decreases and acceleration decreases. As long as acceleration is more than zero, both speed and drag force will increase until the drag force is equal to the weight. At this point, resultant force is zero hence acceleration is zero. Object is moving at terminal velocity.

Concept 3: Graphical Analysis for an Object being dropped with air resistance vs an object dropped without air resistance -- things to take note when drawing the v-t graph:

Graph A: 

• For the object without air resistance, the slope of the graph is a straight line with a constant negative slope because the acceleration due to gravity is constant and directed downwards.

Graph B: 

With air resistance:

• When the object is on an upward journey, the slope is steeper since air resistance is acting in the same direction as weight.
• When the object is on a downward journey, the slope is gentler since the retarding effect of air resistance is in the opposite direction of weight such that net acceleration is less than what it would be due to weight alone.
• When the object is at maximum height, gradient of graph A = gradient of graph B since the only force acting on both balls is weight since the instantaneous velocity is 0.

Concept 4: Graphical Analysis for an Object being dropped with air resistance vs an object dropped without air resistance -- things to take note when drawing the trajectory:

For the graph with air resistance compared to the graph without air resistance:

• It has a shorter range – there is a net deceleration in the horizontal direction and hence a decrease in horizontal speed
• Reduced maximum height – work is done against air resistance, thus the body loses mechanical energy
• Maximum height is displaced to the left – as the object rises, net deceleration is greater than g since air resistance acts in the same direction as weight (as the ball falls, acceleration is smaller than g). Thus speed decreases to zero in a shorter time.
• The path is not symmetrical since horizontal velocity is not uniform.

Concept 5: Projectile Motion on flat ground without air resistance

Underlying principle: Uniform velocity in the horizontal direction and uniform deceleration in the vertical direction.

 Step 1: Resolve the initial velocity into horizontal and vertical components

 Step 2: Find the time taken to reach the maximum height, t

 Step 3: The range is determined by this time 2t since t is the time needed to only travel half the range

Additional Notes:

• To find the velocity of the body before it hits the ground, find:

• To find the angle θ the velocity makes with the ground:

• Condition for maximum range is where the angle of projection = 45°

• Velocity of the body is not zero at maximum height since the horizontal component of velocity is non-zero. Correspondingly, the kinetic energy of the body is not zero.