Calculating the Range and Time of Flight of a Projectile Launched at 30 Degrees

When a ball is projected with a velocity of 10 m/s at an angle of 30 degrees with the horizontal surface, understanding the range and time of flight is crucial. In this article, we will explore the calculations involved in determining these parameters without considering air resistance. Let's delve into the detailed steps and formulas used.

Calculating the Range of a Projectile

The range R of a projectile launched with an initial velocity v_0 at an angle theta; with the horizontal can be calculated using the formula:

R frac{v_0^2 sin(2theta;)}{g}

Here:

v_0 10 , m/s is the initial velocity. theta; 30^circ is the launch angle. g 9.81 , m/s^2 is the acceleration due to gravity.

Step-by-Step Calculation

First, let's check the 2theta; value as sin(2theta;) is required:

2theta; 2 times 30^circ 60^circ sin(60^circ) frac{sqrt{3}}{2}

Substitute the values into the range formula:

R frac{10^2 times frac{sqrt{3}}{2}}{9.81} , m

Calculate the range:

First, calculate sqrt{3} approx 1.732. R approx frac{100 times 1.732}{9.81} , m R approx frac{173.2}{9.81} , m approx 17.65 , m

Upon rechecking, we noticed the range is actually:

R approx frac{50sqrt{3}}{9.81} approx frac{50 times 1.732}{9.81} approx 8.82 , m

Therefore, the range of the projectile is approximately 8.82 , m.

Time of Flight Calculation

The time of flight T can be calculated using the following approach:

T 2 times frac{v_0 sin(theta)}{g}

Here, we can directly substitute the values:

v_0 10 , m/s, , theta 30^circ, , g 9.81 , m/s^2 T 2 times frac{10 sin(30^circ)}{9.81} , s sin(30^circ) 0.5 T 2 times frac{10 times 0.5}{9.81} , s approx 2 times frac{5}{9.81} , s approx 2 times 0.510 , s approx 1.02 , s

The time of flight is approximately 1.02 , s.

Understanding Maximum Height and Rise/Fall Time

The maximum height y_{max} can be calculated using the formula:

y_{max} frac{v_0^2 sin^2(theta)}{2g}

Using the given values:

y_{max} frac{10^2 times (0.5)^2}{2 times 9.81} , m approx frac{25}{19.62} , m approx 1.28 , m

The maximum height is approximately 1.28 , m.

For the time of rise or fall, we use:

t_{rise/fall} frac{v_0 sin(theta)}{g}

Substituting the values:

t_{rise/fall} frac{10 times 0.5}{9.81} , s approx frac{5}{9.81} , s approx 0.510 , s

The time of rise or fall is approximately 0.510 , s.

Refer to the provided screenshot for the detailed motion of the projectile.

Conclusion

In summary, the range of the projectile launched at 30 degrees and 10 m/s is approximately 8.82 , m. The time of flight, accounting for equal rise and fall times, is approximately 1.02 , s. The maximum height reached is around 1.28 , m, and the time of rise or fall is approximately 0.510 , s. These calculations provide a comprehensive understanding of the projectile's motion under the influence of gravity.