Projectile Motion

Optimal Angle for Maximum Range

Projectile motion describes the path of an object launched into the air, subject only to gravity. The angle that maximizes the horizontal distance (range) of a projectile, assuming no air resistance and launch from ground level, is 45°.

• Key Point: The range of a projectile is given by , where is the initial velocity, is the launch angle, and is the acceleration due to gravity.

• Example: Launching a ball at 45° will result in the greatest horizontal distance.

Friction and Circular Motion

Static vs. Kinetic Friction in Curves

Frictional forces are crucial for vehicles navigating curves. Static friction prevents slipping, while kinetic friction acts when sliding occurs.

• Key Point: The maximum static friction force is , where is the coefficient of static friction and is the normal force.

• Example: If Anna's tires have higher static friction than Maria's, Anna can travel faster around the curve without sliding.

Distance and Direction

Vector Addition in Navigation

Traveling in perpendicular directions forms a right triangle, and the total distance from the starting point can be found using the Pythagorean theorem.

• Key Point: If you travel 1 mile north and 1 mile east, your displacement is miles.

• Example: Traveling one mile north, then one mile east, results in a straight-line distance of approximately 1.41 miles from the origin.

Projectile Height and Range Comparisons

Effect of Launch Angle on Maximum Height

For projectiles launched at the same speed, the maximum height depends on the vertical component of velocity.

• Key Point: Maximum height .

• Comparison: A projectile launched vertically () reaches a higher maximum height than one launched at 30°.

Forces and Tension

Tension in a Rope

When two people pull on a rope with equal force, the tension in the rope equals the force applied by each person.

• Key Point: Tension in the rope is equal to the force applied by each person if the forces are equal and opposite.

• Example: If each person pulls with 100 N, the tension is 100 N.

Apparent Weight and Accelerating Systems

Scale Reading in an Accelerating Elevator

The apparent weight of a person in an elevator depends on the acceleration of the elevator.

• Key Point: Apparent weight , where is the elevator's acceleration.

• Example: If a 60 kg person is in an elevator accelerating upward at 2 m/s², the scale reads N.

Inclined Planes and Friction

Angle of Incline and Acceleration

Objects on frictionless inclines accelerate due to gravity's component along the slope.

• Key Point: Acceleration .

• Example: On a 30° incline, m/s².

Circular Motion

Speed and Friction on a Circular Track

For an object moving in a circle, the maximum speed before sliding is determined by static friction.

• Key Point: Maximum speed , where is the radius.

• Example: For and m, m/s.

Collisions and Conservation of Momentum

Mid-Air Collisions

When two objects collide in mid-air, conservation of momentum applies.

• Key Point: Total momentum before collision equals total momentum after, assuming no external forces.

• Example: A bird colliding with a bee will result in both changing velocities according to their masses and initial velocities.

Normal Force on Inclined Planes

Calculating Normal Force

The normal force is the perpendicular contact force exerted by a surface.

• Key Point: On an incline, .

• Example: For a 25 kg sled on a 20° slope, N.

Terminal Velocity and Air Resistance

Terminal Velocity in Free Fall

Terminal velocity is reached when the force of air resistance equals the force of gravity.

• Key Point: For high Reynolds number, air resistance .

• Formula: where is air density, is drag coefficient, is cross-sectional area.

• Example: Tennis ball falling through air reaches a constant speed when drag balances weight.

Vector Addition and Magnitude

Properties of Vector Sums

Vectors can be added using the parallelogram rule. The magnitude of the sum depends on their relative directions.

• Key Point: If vectors and are perpendicular, .

• Example: Two perpendicular vectors of magnitude 3 and 4 have a sum of magnitude 5.

Tables: Forces in Stacked Objects

Force Distribution in Stacked Masses

When multiple objects are stacked, the force on each is determined by the weight of the objects above.

Additional info: The force on the bottom block is the sum of the weights of all blocks above it.

Extra Credit: Spheres and Terminal Velocity

Terminal Velocity Ratio for Spheres

Terminal velocity for spheres falling through air depends on radius and density.

• Key Point: for high Reynolds number.

• Example: Sphere 1 has twice the density and half the radius of Sphere 2; the ratio of their terminal velocities can be calculated using the formula above.

Summary Table: Key Equations

Additional info:

• Some questions require knowledge of vector addition, conservation of momentum, and the effects of friction and air resistance.

• For circular motion, centripetal acceleration is .

• For inclined planes, the angle at which an object starts to slide is .