A ball is thrown with a velocity of 3.0 m/s at an angle of 30o below horizontal. Its vertical velocity is zero when:
The vertical velocity is zero when the ball reaches the highest point of its trajectory. At this point, the ball is momentarily stationary before it starts to fall back down.
Using the given information, we can determine the initial vertical velocity (Vy) using the equation:
Vy = V * sin(theta)
where V is the initial velocity and theta is the angle below the horizontal.
Vy = 3.0 m/s * sin(30o)
Vy = 3.0 m/s * 0.5
Vy = 1.5 m/s
The vertical velocity is zero at the highest point when the ball is momentarily stationary. Therefore, the vertical velocity is zero when Vy = 0 m/s.
0 m/s = 1.5 m/s * sin(theta)
To solve for theta, we can rearrange the equation:
sin(theta) = 0 m/s / 1.5 m/s
sin(theta) = 0
The only angle that has a sine of 0 is 0 degrees. Therefore, the vertical velocity is zero when the angle (theta) below the horizontal is 0 degrees.
To determine when the ball's vertical velocity is zero, we need to consider the motion of the ball in the vertical direction. We can use the equations of motion to analyze the vertical component of the ball's motion.
The vertical motion of the ball can be described by the equation:
v_fy = v_iy + a_y t
Where:
v_fy = final vertical velocity
v_iy = initial vertical velocity
a_y = acceleration due to gravity (approximately -9.8 m/s^2) since it acts downward
t = time
In this case, the initial vertical velocity (v_iy) is given as zero since the ball is thrown at an angle of 30 degrees below the horizontal. This means the initial velocity only has a horizontal component and does not contribute to the vertical motion.
Therefore, the equation simplifies to:
v_fy = a_y t
To find when the vertical velocity is zero, we set v_fy equal to zero and solve for t:
0 = -9.8 t
Rearranging the equation, we get:
t = 0
This means the vertical velocity is zero at the instant when the ball is thrown.
To determine when the ball's vertical velocity is zero, we need to consider the motion of the ball in terms of its horizontal and vertical components.
Given:
Initial velocity (v0) = 3.0 m/s
Launch angle (θ) = 30° (below horizontal)
Vertical velocity (Vy) = 0 (to find)
We know that the motion of an object can be divided into its horizontal and vertical components. The horizontal component is unaffected by gravity, while the vertical component is influenced by gravity.
To find the vertical velocity when it is zero, we can use the following formula:
Vy = V0 * sin(θ) - g * t
where:
Vy is the vertical velocity,
V0 is the initial velocity,
θ is the launch angle,
g is the acceleration due to gravity (approximately 9.8 m/s^2),
t is the time.
Since we are looking for the moment when the vertical velocity is zero, we can set Vy to zero in the equation:
0 = V0 * sin(θ) - g * t
Now, let's solve the equation for t:
g * t = V0 * sin(θ)
t = (V0 * sin(θ)) / g
Substituting the given values:
t = (3.0 m/s * sin(30°)) / 9.8 m/s^2
Using trigonometry, sin(30°) = 0.5:
t = (3.0 m/s * 0.5) / 9.8 m/s^2
t = 0.15 s
Thus, the vertical velocity of the ball is zero at 0.15 seconds after it is thrown.