Physics

Kinematics

Formulas:

s = ut + 1/2 (at^2)
s = 1/2 (v+u)t
v = u + at
v^2 = u^2 + 2as

Definitions

Distance is the length of the path taken. It is a scalar quantity.

Displacement is the distance travelled along a specific direction. It is a vector quantity.

Speed is the rate of change of distance. It is a scalar quantity.

Velocity is the rate of change of displacement. It is a vector quantity.

Acceleration is the rate of change of velocity. It is a vector quantity.

Graphs

The gradient of a displacement-time graph gives the instantaneous velocity at a given point in time.

The gradient of a velocity-time graph gives the instantaneous acceleration at a given point in time, and the area under graph represents the total displacement of an object.

The area under an acceleration-time graph gives the change in velocity of an object.

Free fall

An object is said to be in free fall if the only force acting on it is the gravitational force of the earth. There is an absence of air resistance.

It is assumed to be a constant 9.81m/s^2.

In the presence of air resistances, there is no free fall. Air resistance is a drag force that opposes motion , which is proportional to the speed (slowly-moving) or square of the speed (fast-moving) of the object. The resultant acceleration gradually decreases, until the object reaches terminal velocity.

Projectile motion

A projectile motion follows that of a parabolic curve. An object thrown horizontally experiences a:

- Horizontal motion of constant velocity
- Vertical motion of constant acceleration (9.81m/s^2)

The upwards and side wards forces can be separated into their individual components.

Projectile launched horizontally

Since time of flight depends solely on the vertical component of the projectile,

s = ut +1/2 (at^2) , where s is the initial height reached, u is the initial speed and a is acceleration due to gravity.

The range (maximum horizontal displacement) can be found by,

R = ut , where u is the horizontal speed and t is the time. Horizontal velocity is constant throughout.

Resultant velocity can be found by,

v = √(u)^2 + (at)^2 , where u is the horizontal velocity and at is the vertical velocity at a given point in time.

Relative position of an object can be found by,

Vertical displacement = 1/2 (at^2) , area under graph.
Horizontal displacement = ut , range.

Projectile launched at an angle to the horizontal

By resolving the initial velocity into its horizontal and vertical components, you can get the in terms of u and θ, where u is its initial velocity and θ is the angle from the horizontal.

Initial vertical velocity = usinθ
Initial horizontal velocity = ucosθ

The maximum height reached can be found by,

v^2 = u^2 + 2as, where v is zero, u is the initial vertical velocity and a is the acceleration due to gravity.

Since time of flight depends solely on the vertical component of the projectile,

v = u + at , where v is zero, u is the initial speed, and a is acceleration due to gravity. The time found is the time taken to reach the maximum height, for the time of flight, multiply it by two.

The range (maximum horizontal displacement) can be found by,

R = ut , where u is the horizontal speed and t is the time. Horizontal velocity is constant throughout.

Resultant velocity can be found by,

Instantaneous vertical velocity = initial vertical velocity + at , where acceleration is negative and initial velocity is positive (vice versa).

Let instantaneous vertical velocity be k.

v = √(u)^2 + (k)^2 , where u is the horizontal velocity and k is the vertical velocity at a given point in time.

Air resistance

Air resistance causes the projectile to have a vertical and horizontal drag. The range will be shortened, while the maximum height will be lowered. The horizontal velocity drops over time due to air resistance and the magnitude of the vertical acceleration up and down will be unequal.