To determine the velocity of a projectile after a given time, we need to break down its initial velocity into horizontal and vertical components and then analyze their individual motions.
Given: Initial velocity (v0) = 19.6 m/s Launch angle (θ) = 30° Time (t) = 16 seconds
First, we calculate the horizontal and vertical components of the initial velocity using trigonometry:
Horizontal component (v0x) = v0 * cos(θ) Vertical component (v0y) = v0 * sin(θ)
v0x = 19.6 m/s * cos(30°) ≈ 16.99 m/s v0y = 19.6 m/s * sin(30°) ≈ 9.80 m/s
Since there is no horizontal acceleration, the horizontal velocity remains constant throughout the motion. Therefore, the horizontal component of the velocity at any time remains v0x.
Next, we consider the vertical motion of the projectile. The vertical velocity changes due to the acceleration of gravity (g = 9.8 m/s²) acting in the downward direction. The equation for vertical velocity at a given time is:
v = v0y - g * t
v = 9.80 m/s - (9.8 m/s² * 16 s) ≈ -150.8 m/s
The negative sign indicates that the velocity is directed downward.
Finally, we can combine the horizontal and vertical components to find the resultant velocity of the projectile after 16 seconds using the Pythagorean theorem:
v = √(v0x^2 + v^2)
v = √((16.99 m/s)^2 + (-150.8 m/s)^2) ≈ 151.3 m/s
Therefore, after 16 seconds, the velocity of the projectile will be approximately 151.3 m/s, directed at an angle determined by the arctan(v/v0x):
θ = arctan(v/v0x) ≈ arctan((-150.8 m/s) / (16.99 m/s)) ≈ -85.5°
The negative sign indicates that the velocity is directed downward, which aligns with the vertical component's negative value.