Grade 11 Physical Science: Kinematics - Vectors, Components, Motion, Graphs, Equations, Projectile Motion

Vectors: Revision

• Vector: A physical quantity with both magnitude and direction.

• Scalar: A physical quantity with magnitude only.

• Resultant Vector: A single vector that has the same effect as the original vectors acting together.

• For any vector, both size and direction must be stated.

• Vectors are represented on paper with arrows. The head indicates the direction, and the tail indicates the starting point.

• Vectors should be drawn with a ruler and a pencil.

• The direction and magnitude of the vector should be written in ink alongside the line. Example: 2 m East

• Vectors can be added and subtracted.

• The head-to-tail method is used for vector addition. The second vector starts at the tip of the arrow of the first one.

  • Both vectors need to be drawn to scale.

  • The end result is a triangle, "straight line," or a polygon in the case of many vectors.

• Vectors can be added using Pythagoras or other trigonometric rules.

• A rough diagram needs to be made to illustrate the angles and magnitudes. This does not need to be to scale.

• When two vectors affect the same moving object, both vectors start at the same point. The resultant vector will be the diagonal of the parallelogram formed.

  • The resultant vector also starts at the same point as the other vectors.

  • All the vectors will point AWAY from the starting point.

  • All the vectors must have the same unit and scale.

• A tailwind blows from the BACK of the plane (in the same direction as the plane’s flight), and a headwind blows against the plane’s motion.

Components of Vectors

• A single vector can be broken down into a number of vectors which when added give that original vector.

• These vectors which sum to the original are called components of the original vector.

• The process of breaking a vector into its components is called resolving into components.

• Given vector F, it can be expressed as the vector sum of two perpendicular vectors Fx & Fy.

• $F_x$ is the component of F in the x direction.

• $F_y$ is the component of F in the y direction.

• From trigonometry ratios:

  • $sin\theta = \frac{opposite}{hypotenuse}$

  • $cos\theta = \frac{adjacent}{hypotenuse}$

  • $F_x = F cos \theta$

  • $Fy=Fsin\theta$

Speed, Velocity, Position and Acceleration (Revision)

• Motion: The continuous change in position of an object.

• The position of an object is described relative to a reference point.

• Position is a vector quantity that points from the reference point as the origin.

• Distance: length of path travelled.

• Distance is a scalar quantity.

• Displacement: a change in position.

• Displacement is a vector quantity that points from the initial to the final position.

• Speed: The rate at which distance is covered.

• The speed of a car is not necessarily constant during its trip – the speed usually varies somewhat.

• The average speed is defined as follows:

• The unit of speed is the metre per second ($\frac{m}{s}$).

• The average speed of an object does not give any indication of the different speeds and variation that may take place during shorter time intervals.

• Instantaneous Speed: The speed at a specific moment during motion. It is determined by calculating the average speed for a very short time interval.

• The speed registered on the speedometer of a car is its instantaneous speed.

• Speed is a scalar quantity and is described by magnitude only.

• Velocity is a vector quantity and is described by both magnitude (how much) and direction (which way).

• Average velocity is defined as follows:

  • s is the symbol for displacement (measured in m).

  • t is the symbol for time (measured in s).

  • v is the symbol for average velocity (measured in $\frac{m}{s}$).

• We can change the velocity of an object by:

  • changing its speed, or

  • its direction, or

  • by changing both its speed and direction.

• Motion with changing velocity is called accelerated motion.

• Acceleration simply tells us how quickly the velocity of an object changes.

• Acceleration: the rate of change in velocity. Acceleration is a vector quantity.

• $Acceleration = \frac{change \ in \ velocity}{time \ taken}$

• $a = \frac{\Delta v}{\Delta t}$

• Acceleration is measured in $\frac{m}{s^2}$.

• Another way of representing the equation for acceleration: $a = \frac{v - u}{t}$

  • u for initial velocity

  • v for final velocity

Graphs of Motion (Revision)

• displacement versus time (or distance versus time)

• velocity versus time (or speed versus time)

• acceleration versus time

A. Distance / Displacement / Position vs Time Graphs

• The X axis shows us time and Y axis shows position/ displacement.

• In the graph the position is linearly increasing in positive direction with the time.

• From this linear increasing we can say that velocity is constant / uniform. (If it was not constant we would see a curved line in our graph)

• The gradient of a distance-time graph tells us the speed.

• A steep gradient means a high speed.

• A zero gradient (a horizontal part on the graph) means that the object is not moving.

• If the line is straight (not necessarily horizontal), it means a constant speed.

• A curved line means the object is accelerating or decelerating.

• When the line falls below the 0 mark the displacement is happening in the opposite direction.

• This "negative displacement" should be subtracted from the "positive displacement" to find out what is the resultant displacement and its direction.

• If Position is increasing positive direction: (i.e there is positive acceleration)

  • In this graph velocity is changing.

  • Position does not increase linearly.

  • We can find the velocity of the object from this graph by calculating the gradient of the tangent.

B. Velocity vs Time Graphs

• The slope of the graph is the acceleration.

• The area under the graph is the displacement.

C. Acceleration vs. Time Graphs

• Uniform acceleration occurs when the object accelerates at a constant rate.

• On the acceleration vs time graph of a uniformly accelerating object the acceleration will appear as a horizontal line.

Additional Notes:

• A graph is used to illustrate the mathematical relationship between two quantities.

• Graphs can be useful in describing the motion of a body.

• In graph problems you should be careful while reading it. You can say many things about the motion of the object by just looking of the graph. The important thing is that you must know the relations, meaning of the slopes or area of the graphs..

• Graphs contain a direction. Above the x- axis, the positive position quantities are in the North direction while the negative quantities are in the opposite, South, direction. A position of zero signifies that the object is at the reference point, its starting position.

Distance vs Time Graphs

• The gradient (or slope) of a distance-time graph tells us the speed.

  • A steep gradient means a high speed.

  • A zero gradient (a horizontal part on the graph) means that the object is not moving.

  • If the line is straight (not necessarily horizontal), it means a constant speed.

  • A curved line means the object is accelerating or decelerating.

• When the line falls below the 0 mark the displacement is happening in the opposite direction.

• This "negative displacement" should be subtracted from the "positive displacement" to find out what is the resultant displacement and its direction.

• If Position is increasing positive direction: (i.e there is positive acceleration)

  • In this graph velocity is changing.

  • Position does not increase linearly.

  • We can find the velocity of the object from this graph by calculating the gradient of the tangent.

Velocity vs Time Graphs

• The slope of the graph is the acceleration.

• The area under the graph is the displacement.

• When a velocity vs time graph of an object's motion is a horizontal line the object's acceleration is zero.

Acceleration vs. Time Graphs

• Uniform acceleration occurs when the object accelerates at a constant rate.

• On the acceleration vs time graph of a uniformly accelerating object the acceleration will appear as a horizontal line.

• The following table gives a detailed summary of the mathematical interpretation of graphs of motion

Equations of Motion (Revision)

The four equations of motion where acceleration is constant are:

The list below represents a summary of the physical quantities used in the equations of motion given on the information sheet.

• Note the following points when using equations of motion:

  • Equations of motion apply to linear motion, i.e. motion in a straight line.

  • The positive direction should always be stated because all the quantities except time are vectors. The direction in which the object is initially moving, or about to move, is usually taken as being positive.

  • If an object starts from rest, $v<em>i = 0 \frac{m}{s}$, or if an object is brought to rest $v</em>f = 0 \frac{m}{s}$.

  • If the velocity of an object is increasing, a is positive, and if the velocity of an object is decreasing, a is negative.

  • Most problems dealing with equations of motion will expect you to identify which variables are known and then substitute them into one of the four equations to calculate the unknown value.

  • Method to follow:

    • Start by making a list of the known variables.

    • Then decide which equation you should use to calculate the unknown variable from the known variables.

    • Write down the equation, substitute the values from the problem into it.

    • Use your calculator to work out the unknown quality.

Projectile Motion

• Projectile motion refers to the motion of an object that is thrown, or projected into the air at an angle.

• The vertical motion of a projectile is nothing more than free fall with a constant downward acceleration due to gravity.

• Weight is the gravitational force that the Earth exerts on any object on or near its surface.

• The weight of an object gives you an indication of how strongly the Earth attracts that body towards its centre.

• Weight is calculated as follows: Weight = mg Where m is the mass of the object and g is the acceleration due to gravity.

• A field is a region of space in which a mass experiences a force. Therefore, a gravitational field is a region of space in which a mass experiences a gravitational force.

• Free fall is the term used to describe a special kind of motion in the Earth’s gravitational field.

• Free fall is motion in the Earth’s gravitational field when no other forces act on the object.

• Free fall is basically an ideal situation, since in reality, there is always some air friction which slows down the motion.

• The acceleration due to gravity is constant. This means we can use the equations of motion under constant acceleration (9,8 m.s⁻² downwards) for an object in free fall.

• The path of a projectile is called its trajectory.

• A projectile, once projected, continues in motion by its own inertia and is influenced only by the downward force of gravity.

• An object projected horizontally will reach the ground in the same time as an object dropped vertically.

• No matter how large the horizontal velocity is, the downward pull of gravity is always the same.

Equations of Motion

$a = g = 9.8 \frac{m}{s^2}$
When solving questions it is advisable to do the following:

• Draw a rough sketch

• Write down the given information and what is asked

• Choose up or down as positive direction

• Choose the most appropriate equation

• Use the chosen equation to solve the problem

ABSENCE OF AIR RESISTANCE

• In the absence of air resistance/ air friction a body thrown upwards, downwards or released from rest is in free fall.

• A body that is projected through a gravitational field is known as a projectile

• Now we can consider the motion of objects that are thrown upwards and then fall back to the Earth.

• When an object is thrown straight upwards and then falls straight downwards - this means that there is no horizontal displacement of the object, only a vertical displacement.

• In the absence of air resistance the only force acting on the object is force due to gravity. The object has a constant acceleration (g = 9,8 m.s⁻¹)

• The force of gravity always act towards the centre of the earth.

• When an object if thrown upwards its velocity decreases until it reaches maximum where it momentarily stops and then accelerate downwards.

• In both the upward and downward motion of the acceleration is towards the centre of the earth (downwards)

• When the object is moving upwards the speed decreases until it stops.

• At maximum height the speed is 0 m.s⁻¹

• When the object is moving downwards the speed increases.

• Projectiles take the same the time to reach their greatest height from the point of upward launch as the time they take to fall back to the point of launch.

• Equations of motion are used to solve problems involving vertical projectile motion.

• Remember to choose up or down as positive