Projectile Motion Analysis for Initial Velocity of 20 m/s at 30 Degrees

To analyze the motion of a projectile launched with an initial velocity of 20 m/s at an angle of projection of 30 degrees, we can use standard projectile motion equations. This article will provide the step-by-step solution to determine the range, maximum height, and total time of flight for this scenario.

1. Range R

The range of a projectile is given by the formula:

R frac{v_0^2 sin 2theta}{g}

where:

g is the acceleration due to gravity, approximately 9.81 m/s2 θ is the angle of projection

Substituting the given values:

R frac{20^2 sin 2 times 30^circ}{9.81}

frac{400 times sin 60^circ}{9.81}

frac{400 times frac{sqrt{3}}{2}}{9.81}

frac{200sqrt{3}}{9.81} approx frac{346.41}{9.81} approx 35.32 text{ m}

Therefore, the range R ≈ 35.32 m.

2. Maximum Height H

The maximum height of a projectile is given by the formula:

H frac{v_0^2 sin^2 theta}{2g}

Substituting the given values:

H frac{20^2 sin^2 30^circ}{2 times 9.81}

frac{400 times left(frac{1}{2}right)^2}{19.62}

frac{400 times frac{1}{4}}{19.62}

frac{100}{19.62} approx 5.1 text{ m}

Therefore, the maximum height H ≈ 5.1 m.

3. Total Time of Flight T

The total time of flight for a projectile is given by the formula:

T frac{2v_0 sin theta}{g}

Substituting the given values:

T frac{2 times 20 times sin 30^circ}{9.81}

frac{40 times frac{1}{2}}{9.81}

frac{20}{9.81} approx 2.04 text{ s}

Therefore, the total time of flight T ≈ 2.04 s.

Summary of Results

Range R: ≈ 35.32 m Maximum Height H: ≈ 5.1 m Total Time of Flight T: ≈ 2.04 s

Additional Calculations

Let's explore the vertical and horizontal components of the motion in more detail:

1. Vertical Motion

For vertical motion, we can use the following equation:

v^2 u^2 2as

At the maximum height, the vertical velocity v 0 :

0 (20 sin 30)^2 - 2 times g times s

5 g times s

s frac{5}{g}

If g 10 text{ m/s}^2, s 0.5 text{ m}. If g 9.81 text{ m/s}^2, s approx 0.51 text{ m} (2dp).

This is the maximum height.

For the total time of flight, we use:

s ut frac{1}{2}at^2

0 20 sin 30 t - frac{g}{2} t^2

0 t (10 - frac{g}{2} t)

This gives us t 0 or t frac{20}{g}.

Vertical distance: 0 at the start and when t frac{20}{g}. If g 10, t approx 2 text{ seconds}. If g 9.81, t approx 2.04 text{ seconds} (2dp).

2. Horizontal Motion

For horizontal motion, we use:

s ut frac{1}{2}at^2

where a 0:

s 20 cos 30 t

If t 2 text{ seconds}, s 20 sqrt{3} approx 34.64 text{ m} (2dp).

If t 2.04 text{ seconds}, s approx 35.35 text{ m} (2dp).

This is the range.

Overall, the projectile motion analysis provides detailed insights into the range, maximum height, and total time of flight for a projectile launched with an initial velocity of 20 m/s at 30 degrees.