Determination of Projectile Speed After 3 Seconds

In the study of projectile motion, it's essential to understand how to calculate the speed of the projectile at a specific time. This article will walk you through the process of determining the speed of a projectile launched at 120 meters per second at a 35-degree angle above the horizontal, 3 seconds later, using both traditional and simplified methods. This knowledge is crucial for mastering the principles of projectile motion and enhancing your skills as a Search Engine Optimizer (SEO).

Initial Setup

A projectile is launched at an initial velocity of 120 meters per second at an angle of 35 degrees above the horizontal. We aim to find the projectile's speed after 3 seconds. The key to solving this problem involves breaking down the initial velocity into its horizontal and vertical components and then applying the relevant kinematic equations.

Initial Velocity Components

The initial velocity vector is given as:

v_0 120 , text{m/s}

The launch angle is:

theta 35^circ

The horizontal and vertical components of the velocity can be calculated as follows:

Horizontal Component

[v_x v_0 cdot costheta 120 cdot cos35^circ approx 120 cdot 0.819 98.28 , text{m/s}]

Vertical Component

[v_{0y} v_0 cdot sintheta 120 cdot sin35^circ approx 120 cdot 0.573 68.76 , text{m/s}]

Vertical Velocity After 3 Seconds

The vertical component of the velocity changes due to the influence of gravity. The vertical velocity after 3 seconds can be calculated using the following equation:

[v_y v_{0y} - g cdot t]

where g 9.81 , text{m/s}^2 is the acceleration due to gravity and t 3 , text{s}.

[v_y 68.76 - 9.81 cdot 3 68.76 - 29.43 approx 39.33 , text{m/s}]

Horizontal Velocity

Since there is no horizontal acceleration, the horizontal velocity remains constant:

[v_x 98.28 , text{m/s}]

Resultant Speed

The resultant speed of the projectile can be calculated using the Pythagorean theorem:

[v sqrt{v_x^2 v_y^2}]

Substituting the values:

[v approx sqrt{98.28^2 39.33^2} approx sqrt{9665.51 1545.86} approx sqrt{11211.37} approx 105.87 , text{m/s}]

Simplified Method

A more efficient approach uses the derived formula:

[v sqrt{[u cdot cosalpha]^2 [u cdot sinalpha - g cdot t]^2}]

Substituting the given values:

[u 120 , text{m/s}, , alpha 35^circ, , g 9.81 , text{m/s}^2, , t 3 , text{s}]

[v sqrt{[120 cdot cos35^circ]^2 [120 cdot sin35^circ - 9.81 cdot 3]^2}]

[v approx sqrt{[98.28]^2 [68.76 - 29.43]^2} approx sqrt{9665.51 1545.86} approx 105.9 , text{m/s}]

Conclusion

This method provides a straightforward way to determine the projectile's speed after any given time without the need for repeated calculations. Understanding these principles not only aids in solving complex problems but also enhances your SEO skills, making you better equipped to optimize content for search engines.

Keywords

projectile motion, velocity components, kinematic equations