Recollection Guide

1. Scalars and vectors

• Scalars are quantities that have only magnitude, such as mass and temperature.
• Vectors are quantities that have both magnitude and direction, such as force and velocity.

2. Vector addition and subtraction

• Vectors can be added and subtracted by using the parallelogram law.
• To add two vectors, place them head-to-tail and draw a parallelogram between them. The vector that goes from the tail of the first vector to the head of the second vector is the sum of the two vectors.
• To subtract a vector from another vector, place them head-to-tail and draw a parallelogram between them. The vector that goes from the head of the first vector to the tail of the second vector is the difference of the two vectors.

3. Dot product and cross product of vectors

• The dot product of two vectors is a scalar quantity that is equal to the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them.
• The cross product of two vectors is a vector quantity that is perpendicular to both of the two vectors and has a magnitude equal to the product of the magnitudes of the two vectors multiplied by the sine of the angle between them.

4. Unit vectors

• Unit vectors are vectors that have a magnitude of 1.
• Unit vectors are often used to represent the directions of vectors.

5. Position, displacement, velocity, and acceleration

• Position is the location of an object at a given time.
• Displacement is the change in position of an object over a given time interval.
• Velocity is the rate of change of an object’s position with respect to time.
• Acceleration is the rate of change of an object’s velocity with respect to time.

6. Equations of motion

• The equations of motion are three mathematical equations that describe the motion of an object in a straight line.
• The first equation of motion is:

v = u + at

• where v is the final velocity of the object, u is the initial velocity of the object, a is the acceleration of the object, and t is the time interval.
• The second equation of motion is:

s = ut + 1/2at^2

• where s is the displacement of the object, u is the initial velocity of the object, a is the acceleration of the object, and t is the time interval.
• The third equation of motion is:

v^2 = u^2 + 2as

• where v is the final velocity of the object, u is the initial velocity of the object, a is the acceleration of the object, and s is the displacement of the object.

7. Projectile motion

• Projectile motion is the motion of an object that is launched into the air, such as a ball or a rocket.
• The equations of motion for projectile motion are:

x = u_xt

y = u_yt - 1/2gt^2

• where x is the horizontal displacement of the object, u_x is the initial horizontal velocity, t is the time interval, y is the vertical displacement of the object, u_y is the initial velocity of the object, g is the acceleration due to gravity, and t is the time interval.

8. Circular Motion

• Circular motion is the motion of an object that moves in a circular path, such as a planet orbiting the sun or a car going around a curve.
• The equations of motion for circular motion are:

v = ωr

a = ω^2r

• where v is the tangential velocity of the object, ω is the angular velocity, r is the radius of the circular path, and a is the centripetal acceleration.

9. Relative velocity and acceleration

• Relative velocity is the velocity of an object relative to another object.
• Relative acceleration is the acceleration of an object relative to another object.

10. Angular motion

• Angular motion is the rotation of an object around a fixed point.
• The equations of angular motion are:

ω = Δθ / Δt

α = Δω / Δt

• where ω is the angular velocity of the object, θ is the angle that the object has rotated through, t is the time interval, and α is the angular acceleration of the object.

11. Uniform circular motion

• Uniform circular motion is circular motion in which the object moves with a constant angular velocity.
• The equations of uniform circular motion are:

v = ωr

a = ω^2r

• where v is the tangential velocity of the object, ω is the angular velocity, r is the radius of the circular path, and a is the centripetal acceleration.