Projectile and Circular Motion

Projectile Motion

Projectile motion is a form of motion in which an object or particle (called a projectile) is thrown near the earth's surface, and it moves along a curved path under the action of gravity only. The only force of significance that acts on the object is gravity, which acts downward to cause a downward acceleration.

To find Vo_y , Vo_x and time use this equation : (Y₀ is initial height)

Vo_y = Vo * Sin Θ                Vo_x = Vo * Cos Θ                0 = Y₀ + Vo_y * t + 0.5 * g * t²

To find the Magnitude of projectile motion use this equation :

Vo = √(Vo_y² + Vo_x²)

To find the Direction of projectile motion use this equation :

Tan Θ = Vo_y ÷ Vo_x

To find the max range of projectile motion use this equation :

X = Vo_x * time

To find the max height of projectile motion use this equation :

X_H =  Vo_y² / 2g


Examples : 

if Vo = 10 and Θ = 30 then first we must find the time by using Vo_y / g which is 10 * Sin 30 = 5 / 10, but it is only the time to reach the max height thus 0.5 * 2 is the time for max range which is 1 second.
Now to find the max range get the Vo_x by 10 * Cos 30 = 5√3 and times it with the time which is 1 second so the max range is 5√3

Circular Motion

Circular Motion is a movement of object along side of the circumference of a circle or rotation along a circular path.

In Circular Motion V is changed to Ω which is omega while a = α which is alpha.

Equation used in Circular Motion :

V = Ω * R (R is radius) this is used to describe relation between angular velocity and linear velocity

a = α * R (in rad/s²) this is used to describe relation between angular accel and linear accel

V_1 = V₀ + a * t (in m/s)

Ω_{1  =} Ω₀ + α * t (in rad/s)

a_c = V² / R (in rad/s²)

α_c = Ω² / R (in rad/s²)

F_c = M * α_c (in newton) this is used to find centripetal force

V = √(µ_s * g * R) (µ_s dry = 0.9 ice = 0.1) this is used to find velocity needed to turn on cliff

V = √((G * M_earth) / R) this is used to find velocity of satellites revolving the earth


Converting cycle = ° = rad

1 cycle = 360° = 2π rad

Examples : 

Convert to radian : 
180° = 180 / 360 * 2π rad = π rad
90° = 90 / 360 * 2π rad = π / 2 rad

Convert to ° :
π / 4 rad = (π / 4) / 2π * 360 = 45°
3π / 4 rad = (3π / 4) / 2π * 360 = 135°


Circular Motion in 2 Wheels

For the first picture V₁ = V₂ while the second picture  Ω₁ = Ω₂ 

Examples : 

The first picture if R1 = 10, R2 = 20 and  Ω₁ = 1 then ? = Ω₂

To solve this use the equation V = Ω * R. First find the V which is 1 * 10 = 10, then using the same equation 10 = Ω₂ * 20 which is 10 / 20 = 0.5 rad/s

The second picture if R1 = 10, R2 = 20 and V₁ = 10 then ? = V₂ 

To solve this use the equation V = Ω * R. First find the Ω which is 10 / 10 = 1, then using the same equation V₂ = 1 * 20 which is 20 m/s