Projectile Motion: Calculating the Range of a Body Thrown at an Angle

In physics, projectile motion involves the motion of an object thrown or projected into the air, subject to only the acceleration due to gravity. Understanding the principles of projectile motion allows us to determine the range and trajectory of a projectile. This article delves into how to calculate the range of a body projected from the ground at an initial speed of 80 m/s at an angle of 30°.

Given Data

Initial speed v0 80 m/s Angle of projection theta; 30° Acceleration due to gravity g 9.81 m/s2

Step-by-Step Calculation

Step 1: Resolve the Initial Velocity into Horizontal and Vertical Components

The initial velocity can be resolved into horizontal and vertical components, which are the horizontal and vertical velocity components at the start of the motion. The horizontal component of velocity vx and the vertical component of velocity vy are calculated as follows:

Horizontal component of velocity:

vx v0 middot; costheta; 80 middot; cos(30°) 80 middot; 0.866 69.28 m/s

Vertical component of velocity:

vy v0 middot; sintheta; 80 middot; sin(30°) 80 middot; 0.5 40 m/s

These components help us understand how the projectile will move horizontally and vertically over time.

Step 2: Calculate the Time of Flight

The total time of flight of the projectile can be calculated using the vertical motion of the object. The formula for the time of flight is:

T 2 middot; frac;vy{g} 2 middot; frac;40{9.81} 8.16 seconds

Step 3: Calculate the Horizontal Distance Traveled (Range)

The horizontal distance traveled or the range R can be calculated using the formula for range:

R vx middot; T 69.28 middot; 8.16 565.07 meters

Therefore, the projectile will strike approximately 565.07 meters from the starting point.

Alternative Calculation Methods

Another common way to solve the range of a projectile is by using the simplified formula:

Range u2 middot; sin(2theta;) middot; frac;1{g}

Where u is the initial speed, theta; is the angle of projection, and g is the acceleration due to gravity. Plugging in the given values:

Range (802) middot; sin(60°) middot; frac;1{9.81} 6400 middot; 0.866 middot; frac;1{9.81} 565 meters

The calculation confirms that the projectile will strike at a point 565 meters from the starting point.

Conclusion

The problem of determining the range of a body projected upward from the ground at an angle of 30° with an initial speed of 80 m/s can be effectively solved using the principles of projectile motion. Through step-by-step calculations, we can determine the time of flight and the horizontal distance (range) traveled by the projectile.

Understanding these calculations not only enhances your mathematical and physical skills but also provides insights into the behavior of projectiles in real-world scenarios, such as sports, engineering, and astronomy.