Study guide: Grade 11 Projectile and Equilibrium Physics

📋 Course Outline

1. Velocity Vectors and Components
2. Relative Velocity and River Crossing
3. Horizontal Projectile Motion
4. Angled Projectile Motion
5. Moment of a Force
6. Equilibrium of Forces

📖 1. Velocity Vectors and Components

🔑 Key Concepts & Definitions

• Velocity vector: has both magnitude and direction

★ Must-know

📐 Formula — A velocity of magnitude $v$ at angle $\theta$ to the horizontal has components $v_x=v\cos\theta$ and $v_y=v\sin\theta$.

Further detail

• Two perpendicular velocity components of 20 m/s have a resultant magnitude of 28.3 m/s.

💡 Memory Hook

Picture perpendicular arrows forming a right triangle: horizontal, vertical, resultant.

📖 2. Relative Velocity and River Crossing

📝 Essential Points

📐 Formula — The velocity of B relative to A is $\vec v_{B/A}=\vec v_B-\vec v_A$.

• For a boat aimed perpendicular to a river bank, the crossing component determines time, while the current determines downstream drift.

📐 Formula — The resultant speed of perpendicular velocities $v_1$ and $v_2$ is $v=\sqrt{v_1^2+v_2^2}$.

📐 Formula — To cross a river of width $w$ with perpendicular speed $v_\perp$, the crossing time is $t=\frac{w}{v_\perp}$ and the downstream drift is $d=v_ct$.

📐 Formula — For a boat to cross straight across with no downstream drift, its upstream velocity component must equal the current speed: $v\sin\theta=v_c$.

💡 Memory Hook

Ground velocity combines motion through the medium with medium motion; relative velocity subtracts observer motion.

📖 3. Horizontal Projectile Motion

🔑 Key Concepts & Definitions

• Horizontal projectile: is launched with horizontal velocity and zero initial vertical velocity

★ Must-know

📐 Formula — For a horizontal projectile launched from height $h$, the flight time is $t=\sqrt{\frac{2h}{g}}$.

📐 Formula — The horizontal range of a horizontal projectile is $R=v_0\sqrt{\frac{2h}{g}}$.

• The horizontal velocity of a horizontal projectile remains constant while its vertical velocity increases in magnitude as $v_y=gt$.

Further detail

• A package released from a level-flying plane follows a parabolic path relative to the ground.

📐 Formula — The impact speed of a horizontal projectile is $v=\sqrt{v_x^2+v_y^2}$.

💡 Memory Hook

Horizontal motion stays uniform while gravity causes vertical acceleration, producing a parabolic path.

📖 4. Angled Projectile Motion

★ Must-know

📐 Formula — For a projectile launched at speed $v_0$ and angle $\theta$ on level ground, the total flight time is $T=\frac{2v_0\sin\theta}{g}$.

📐 Formula — The time to reach maximum height is $t_{max}=\frac{v_0\sin\theta}{g}$.

📐 Formula — The maximum height of a level-ground projectile is $H=\frac{v_0^2\sin^2\theta}{2g}$.

• For a fixed initial speed and level ground, maximum range occurs at a launch angle of 45°.

Further detail

• For complementary launch angles with the same speed, the ranges are equal, and the steeper angle produces the greater maximum height.

📐 Formula — For a projectile on level ground, the relation between maximum height, range, and launch angle is $\tan\theta=\frac{4H}{R}$.

💡 Memory Hook

Launch components → rise → highest point → fall → landing.

📖 5. Moment of a Force

🔑 Key Concepts & Definitions

• Moment of a force: the turning effect of the force about a pivot

★ Must-know

📐 Formula — The moment of a force is $\tau=Fd$, where $d$ is the perpendicular distance from the pivot to the line of action.

📐 Formula — For a force $F$ applied at angle $\theta$ to a lever of length $L$, the moment is $\tau=FL\sin\theta$.

📌 A force whose line of action passes through the pivot has zero moment about that pivot.

Further detail

• The SI unit of moment is the newton metre (N m), and a counterclockwise moment is usually positive.

• A 5.0 N force applied perpendicularly 0.40 m from a pivot produces a moment of 2.0 N m.

💡 Memory Hook

A force through the pivot has zero moment, whereas a perpendicular force away from the pivot produces torque.

📖 6. Equilibrium of Forces

🔑 Key Concepts & Definitions

• Translational equilibrium: when the vector sum of all forces is zero

★ Must-know

📌 A body is in rotational equilibrium when the algebraic sum of its moments about a pivot is zero.

📐 Formula — For a body in translational equilibrium, $\sum \vec F=0$.

• 🔄 A cable-equilibrium problem is solved by these steps:
  1. Resolve each tension into horizontal and vertical components
  2. Set the horizontal force sum to zero
  3. Set the vertical force sum to zero
  4. Solve the equations with the weight

Further detail

• For a sign hanging at rest, the upward vertical components of the cable tensions balance its weight, while opposing horizontal components balance each other.

💡 Memory Hook

Translational equilibrium requires zero net force, while rotational equilibrium requires balanced moments.

📊 Synthesis Tables

Projectile Motion Comparison

Feature	Horizontal projectile	Angled projectile
Initial vertical velocity	Zero	$v_0\sin\theta$
Flight time	$\sqrt{2h/g}$	$2v_0\sin\theta/g$
Horizontal velocity	Constant launch speed	Constant component $v_0\cos\theta$