Motion: Uniform Acceleration and Vectors

Equations of Motion

• Useful formulas can be used when an object moves at a uniform acceleration.
• The variables:
  • s = displacement
  • u = initial velocity
  • v = final velocity
  • a = acceleration
  • t = time
• The formulas:
  • v = u + at
  • s = ((u+v)/2)t
  • s = ut + (1/2)at²
  • v² = u² + 2as
• In approaching questions, writing out the values is useful to determine the best formula to use.
• Example: A stone is dropped from a bridge 50 m above the water. To find the final velocity (v) and time (t):
  • s = 50 m
  • u = 0 m/s
  • v = ?
  • a = 9.81 m/s²
  • t = ?
• Because it's dropped, initial velocity is zero; acceleration is gravity (9.81 m/s²).
• Solve for v (31.3 m/s) using v² = u² + 2as.
• Then solve for t (3.19 s) using s = ut + (1/2)at².

Displacement, Velocity, and Acceleration-Time Graphs

• Distance: A scalar quantity that describes how much ground an object has covered.
• Displacement (s): A vector quantity that describes the overall distance from the starting position in a certain direction.
• Speed: A scalar quantity that describes the distance traveled per unit time.
• Velocity (v): Is the rate of change of displacement (Δs/Δt).
• Acceleration (a): Is the rate of change of velocity (Δv/Δt).
• Uniform acceleration is where the acceleration of an object is constant.
• Acceleration-time graphs: The area under the graph represents the change in velocity.
• Velocity-time graphs: Represent the change in velocity over time.
  • The gradient illustrates acceleration.
  • The area calculates displacement.
• Displacement-time graphs' gradient equals velocity, showing the change in displacement over time.
• Instantaneous velocity is velocity at a specific moment, found by the gradient of a tangent on a displacement-time graph.
• Average velocity is the velocity over a specified time frame, which is the final displacement divided by time.

Scalars and Vectors

• Scalars are physical quantities that have a magnitude.
• Vectors describe magnitude and direction.
• Examples:
  • Scalars
  • Distance
  • Speed
  • Mass
  • Temperature
  • Vectors
  • Displacement
  • Velocity
  • Force/Weight
  • Acceleration
• Always specify direction when giving a vector as a solution to a problem.
• Vector notation: bold letter (a), underlined letter (a), or letter with an arrow above it:
  • bold letter is typically used when typing
  • underlined and arrow are used in written algebra

Resolving Vectors

• Resolving a vector splits it into vertical and horizontal components.
• Methods:
  • Calculation: Uses trigonometry. Split a vector V into components x and y:
    • x = V cos θ
    • y = V sin θ
  • Hint:
    • Moving from the original vector through the angle θ to the component, use cos.
    • Moving away from the angle θ to the component, use sin.
  • Scale drawing:
    • Choose an appropriate scale and note it.
    • Use a ruler and protractor to draw the vector V. Draw horizontal and vertical components.
    • Measure the length of the components and find their value using the scale.

Adding Vectors

• Vector addition methods:
  • Calculation: For perpendicular vectors, use:
    • Pythagoras' theorem to find the magnitude.
    • Trigonometry to find the direction. Resultant vector 5 N θ 12 N 12 N
  • The angle found must qualify the direction
  • Scale drawing: Use for vectors at angles other than 90°.
    • Draw a scale diagram with a ruler and protractor, and note the scale. N Scale: 1cm = 10 m 2 cm 3 cm Displacement 60° θ
    • Measure and convert to find displacement magnitude and direction.
• Add/subtract vectors moving in the exact same direction, otherwise resolve them into components.

Projectile Motion

• Vertical and horizontal components of a projectile's motion are independent
• Projectiles can be evaluated separately using the uniform acceleration formula
• Acceleration is constant

Free-Body Force Diagrams

• Illustrates all forces acting on an object.
• Shows how forces compare, indicating motion, such as constant velocity if forces balance: Air resistance Normal reaction force Weight Driving force

Newton's Laws of Motion

• Newton's 1st law: An object remains at rest or constant velocity unless a resultant force acts on it.
• Newton's 2nd law: Acceleration is proportional to the resultant force experienced by the object (F = ma).
  • Mass must be constant
  • Can derive Newton's first law by substituting a resultant force of 0 N, which will result in an acceleration of 0 m/s.
• Terminal velocity: This occurs when frictional and driving forces are equal, having no resultant force, thus constant velocity.
  • Example: A skydiver accelerates until air resistance equals weight.

Gravitational Field Strength and Weight

• Gravitational field strength (g): Is the force per unit mass in a gravitational field (g = F/m).
  • F is the force and m is the mass
• Weight (W): Is the gravitational force on an object due to its mass (W = mg).
  • g = gravitational field strength.

Newton's Third Law of Motion

• For each force experienced by an object, the object exerts an equal and opposite force.
• Example: A book on a table exerts weight, and the table exerts an equal normal reaction force, so the book stays stationary.

Momentum

• Momentum (p): Is the product of mass and velocity (p = mv).

Principle of Conservation of Linear Momentum

• In any interaction, momentum always remains constant if no outside forces exist.
• The momentum before equals the momentum after an event or collision.
• Newton's second law: (F = Δ(mv)/Δt), connecting force to the rate of momentum change.
• Newton's third law: states that momentum change sums to zero in a system of reaction.

Moments

• The moment of a force is force multiplied by perpendicular distance from the force's line of action - Moment = Force × Perpendicular distance.
• Can also be expressed as Moment = Fx, where x is the perpendicular distance.

Centre of Gravity and Principle of Moments

• Principle of moments: An object in equilibrium has equal clockwise and anticlockwise moments around a pivot.
• The centre of gravity represents the point where gravity appears to act within an object.
• Finding the centre of gravity:
  • Uniform objects have their centre of gravity right at the centre
  • Irregular objects can have their centre of gravity found with:
    • A plumbline and attaching an object to it from different points.
    • The centre of gravity aligns where all three lines intersect.
• Weight of an object acts at its centre of gravity and is vital for calculating moments.

Work

• Work done (W) is defined as the force causing a motion multiplied by the distance traveled in the direction of the motion (W = FΔs).
  • F is the force, Δs is the distance
• If the force isn't in the direction of motion, resolve it: W = Fs cos θ.

Kinetic Energy

• Kinetic energy (Ek) is the energy of an object due to its motion (Ek = (1/2)mv²).
  • m = mass
  • v = velocity

Gravitational Potential Energy

• Gravitational potential energy (Egrav) is energy due to position in a gravitational field.
• The change in gravitational potential energy near Earth is: ΔEp = mgΔh.

Principle of Conservation of Energy

• Energy cannot be created or destroyed, only transferred; total energy in a closed system is constant (Total energy in = Total energy out).
• Work can be done by the ball to work against resistive forces. This means the initial kinetic energy is not equal to the maximum gravitational potential.

Power

• Power (P) is the rate of energy transfer:
• P = E/t
• P = W/t
• Work done by electrical appliance: Energy transferred = P × △t

Efficiency

• Efficiency is the measure of how efficiently energy is transferred by a system.
• Efficiency = useful power output / total power input OR useful energy output / total energy input.

Explore motion concepts, including uniform acceleration and its equations. Learn about scalar and vector quantities, Newton's laws of motion, and terminal velocity. Understand vector resolution and graphical representations of motion.