Answer:
12.369 m/s.
Step-by-step explanation:
To find the final velocity of the doll as it hits the ground, we need to break down the initial velocity into its horizontal and vertical components.
Given:
Initial velocity (magnitude): 6.0 m/s
Angle above the horizontal: 32°
Vertical displacement: 5.4 m
First, let's find the horizontal component of the initial velocity (Vx). We can use trigonometry to determine this:
Vx = initial velocity * cos(angle)
Vx = 6.0 m/s * cos(32°)
Vx ≈ 5.091 m/s
Next, let's find the vertical component of the initial velocity (Vy):
Vy = initial velocity * sin(angle)
Vy = 6.0 m/s * sin(32°)
Vy ≈ 3.165 m/s
Now we can analyze the vertical motion of the doll. The vertical displacement (Δy) is given as 5.4 m. We can use the following equation of motion to find the time taken to reach the ground:
Δy = Vy * t + (1/2) * g * t^2
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
5.4 m = 3.165 m/s * t + (1/2) * 9.8 m/s^2 * t^2
Simplifying the equation, we have:
4.9t^2 + 3.165t - 5.4 = 0
Using the quadratic formula, we can solve for t:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4.9, b = 3.165, and c = -5.4. Plugging in these values, we get:
t ≈ 0.956 s or t ≈ -1.156 s
Since time cannot be negative in this context, we discard the negative value and consider t ≈ 0.956 s as the time taken to reach the ground.
Finally, we can find the final vertical velocity (Vfy) using the equation:
Vfy = Vy + g * t
Vfy = 3.165 m/s + 9.8 m/s^2 * 0.956 s
Vfy ≈ 12.369 m/s
Therefore, the final velocity of the doll as it hits the ground is approximately 12.369 m/s.