Dynamics of a Horizontally Projected Ball: Speed and Trajectory Analysis

When a ball is projected horizontally from the top of a cliff, its motion can be analyzed by breaking it down into horizontal and vertical components. Given no air resistance, the horizontal velocity remains constant while the ball accelerates downwards due to gravity. This tutorial will walk you through the detailed analysis of the ball's speed at a specific time point, illustrating the principles of motion and their application.

Horizontal Motion

The ball's horizontal motion is straightforward because there is no force acting on it to change its horizontal velocity. This means the horizontal speed v_x remains unchanged and is given as:

v_x 40 m/s

Vertical Motion

The vertical motion of the ball is influenced solely by the acceleration due to gravity, g, which is approximately 9.81 m/s2. The vertical velocity v_y after time t can be calculated using the formula:

v_y g middot; t

For t 3 s, we have:

v_y 9.81 m/s2 middot; 3 s 29.43 m/s

Resultant Speed

The total speed of the ball after 3 seconds is found using the Pythagorean theorem, since the horizontal and vertical motions are perpendicular to each other. Therefore, the resultant speed v is:

v sqrt{(v_x sup2; v_y sup2;)}

Substituting the values we get:

v sqrt{(40 m/s2 29.43 m/s2) sqrt{1600 867.84} sqrt{2467.84} ≈ 49.67 m/s}

Conclusion and Analysis

Using vector addition, the speed at 3 seconds can be determined. The horizontal and vertical components of the velocity are:

V_x 40 m/s (horizontal)

V_y -29.43 m/s (vertical, negative as it is downward)

The resultant speed is:

V sqrt{(40 m/s2 29.43 m/s2) 49.67 m/s}

The angle below the horizontal is approximately 36 degrees, confirming the speed calculation.

Vertical and Horizontal Velocities After 3 Seconds

After 3 seconds, the vertical velocity is:

V_y -g middot; t -29.43 m/s

And the horizontal velocity remains unchanged:

V_x 40 m/s

Vector Addition for Combined Velocity

To find the combined velocity, we perform vector addition. In this case, the velocities form a right-angled triangle with sides of 30 m/s (vertical) and 40 m/s (horizontal). The hypotenuse (total speed) is 50 m/s, confirming our earlier calculation.

Conclusion

In summary, the speed of the ball 3 seconds after being projected horizontally is approximately 49.67 m/s. This value is derived from the combination of constant horizontal velocity and vertical velocity influenced only by gravity. Understanding this analysis is crucial for grasping the dynamics of projectile motion under the influence of gravity.