Understanding Motion: Distance, Displacement, Velocity

• Motion description depends on the frame of reference.
• Distance refers to the length an object moves along its path.
• Displacement is the difference between the object's initial and final positions.
• Speed is the rate at which an object changes location and is a scalar quantity.
• Velocity describes the speed and direction of an object and is a vector quantity.
• Average speed is the distance travelled divided by the time taken.
• Average velocity is displacement divided by time.
• Position vs. time graphs are used to analyse motion.
• The slope of a position vs time graph indicates velocity.
• The slope of a velocity vs. time graph indicates acceleration.
• The area under a velocity vs. time curve indicates displacement.
• Acceleration is the rate of change of velocity.
• Average acceleration is the change in velocity divided by the time it takes for the change to occur.
• Acceleration is a vector quantity because it has both magnitude and direction.
• Kinematic equations relate time, displacement, velocity, and acceleration for objects in motion.
• These equations apply when acceleration is constant.
• Motion description depends on the reference frame from which it is described.
• Distance refers to the length of the path an object moves along.
• Displacement is the difference in the initial and final positions of an object.

Vectors

• Velocity is a vector quantity that describes the speed and direction of an object.
• Average velocity is displacement over the time period in which the displacement occurs.

Graphs

• The slope of a position vs. time graph is the velocity.
• The slope of a velocity vs. time graph is the acceleration.
• The area under a velocity vs. time curve is the displacement.

Kinematics

• These equations show how time, displacement, velocity, and acceleration are related for objects in motion.
• In 1D motion, direction is indicated by a plus or minus sign, whereas in 2D motion, a vector describes motion in two perpendicular directions, such as vertical and horizontal.
• 2D vectors are made up of vertical and horizontal components, whereas 1D motion can be treated mathematically the same as scalars.
• In a 1D problem, one of the components has a value of zero.
• For 2D vectors, a frame of reference such as a coordinate system is used.
• 2D displacement may require the use of the Pythagorean theorem and trigonometric identities to find resultant magnitude and direction.
• Distance is the length of the path an object moves and is a scalar.
• A scalar is a quantity that has magnitude, or size, but not direction.
• Displacement is the change in position of an object from a fixed axis.
• Displacement is the difference between the final and initial positions of an object and is a vector.
• A vector is a quantity that has both magnitude and direction.

Distance vs Displacement

• Distance is the measurement of the total length of your path from the starting position to the final position
• Displacement is the measurement from the initial position to your final position
• Distance describes the total path between starting and ending points
• Displacement describes the shortest path between starting and ending points
• One-dimensional motion cannot have zero distance with a nonzero displacement.
• One-dimensional motion can have zero displacement with nonzero distance because distance has only magnitude and any motion will be the distance it moves.
• When calculating displacement, the direction matters, but when calculating distance, the direction does not matter.

Scalars vs Vectors

• Scalar: A scalar is a quantity that is fully described by its magnitude, or size, only
• Distance and speed are examples of scalar quantities.
• Vector: A vector is a quantity that is fully described by both its magnitude and direction.
• Displacement and velocity are examples of vector quantities.

Right Triangles

• For a right triangle, the sine, cosine, and tangent of an angle e are defined in relation to the adjacent side, the opposite side, and the hypotenuse:
• Sine: sin(0) = Opposite / Hypotenuse
• Cosine: cos(0) = Adjacent / Hypotenuse
• Tangent: tan(0) = Opposite / Adjacent
• A vector can be broken down into its x and y components using trigonometry.

Velocity vs Time

• For a velocity vs. time graph, the area under the curve represents the displacement.
• Calculating displacement from a velocity vs. time graph can be achieved by estimating shapes over smaller intervals to find the areas.
• When the graph is a curve, you may need to estimate the area.
• For a straight-line graph, calculations are simple, but for an undefined curve, estimating shapes over smaller intervals is necessary to find the areas.
• The graph of position versus time is a curve rather than a straight line when velocity isn't constant.
• The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time.
• The slope at any point on a position-versus-time graph is the instantaneous velocity at that point.
• It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line.
• If the velocity is not constant, the position is a curve.
• Taking the instantaneous velocity at each point on this graph will result in a straight line with a positive slope.
• Taking the area under the slope gives the displacement.
• When dealing with a curved velocity vs time graph, finding the area which relates to displacement, requires approximation by dividing the graph into sections.
• The instantaneous acceleration at a given time can be found by drawing a tangent line at that instant and calculating its slope.

Projectile Motion

• Projectile motion is the motion of an object thrown (projected) into the air
• After the initial force that launches the object, it only experiences the force of gravity.
• The path an object follows under projectile motion is its trajectory.
• Air resistance is ignored to simplify calculations in introductory physics.
• The acceleration is constant and due to gravity, approximately 9.80 m/s² near the Earth's surface.
• The kinematic equations are used to describe the motion.
• At the maximum height of the trajectory, the vertical velocity is zero.
• In projectile motion, horizontal and vertical motions are independent, meaning that they don't influence one another.
• Vertical motion is affected by gravity, while horizontal motion remains constant (if air resistance is ignored).
• Kinematics is the study of motion without considering its causes, and it involves objects in motion.

Distance vs Displacement

• Distance: The length of the path an object moves along.
• Displacement: The difference between the initial and final positions of an object.
• Average Speed: Scalar quantity that is distance travelled divided by the time during the motion.
• Velocity: Vector quantity that describes the speed and direction of an object.
• Graphs can reveal relationships between physical quantities.

Acceleration

Velocity vs Time graph

• Acceleration is the rate of change of velocity
• The slope of a velocity vs. time graph is the acceleration.
• Kinematics is the study of motion without regard to its causes.
• Describing motion requires a reference frame, which is a fixed point or group of points chosen for observation. The choice of reference frame affects how motion is perceived and described.
• Distance is the total length of the path travelled by an object.
• Displacement is the change in position of an object, calculated as the final position minus the initial position.

Speed vs Velocity

• Average speed is the distance travelled divided by the time taken.
• Average velocity is the displacement divided by the time taken. It is a vector quantity.
• Instantaneous speed is the speed at a specific instant in time.
• Instantaneous velocity is the velocity at a specific instant in time.
• Graphs are used to represent motion, with the horizontal axis typically representing the independent variable (time) and the vertical axis representing the dependent variable (position).
• The slope of a position versus time graph represents velocity.
• The Sl unit of force is the newton (N), which is the force needed to accelerate a 1-kg system at the rate of 1 m/s².
• Acceleration is the rate at which velocity changes; it is a vector quantity.
• Average acceleration is the change in velocity divided by the elapsed time.
• Instantaneous acceleration is the acceleration at a specific point in time.
• An object can accelerate in three ways: Speeding up, slowing down, changing direction.
• When problem-solving, it is important to check the answer to see if it is reasonable by assessing its magnitude, sign, and units.
• Time is customarily plotted on the x-axis because it is the independent variable.
• The slope of a velocity vs. time graph is the acceleration.