A boy wants to throw a ball into his friend’s window 16.0 m above. Assuming it just reaches the window, he throws the ball at 55.0° to the ground. At what velocity should he throw the ball?
Range = Vo^2*sin(2A)/g = 16 m.
Vo^2*sin(110)/9.8 = 16
0.09589Vo^2 = 16
Vo^2 = 166.863
Vo = 12.92 m/s.
To find the velocity at which the boy should throw the ball, we can use the principles of projectile motion.
First, let's break down the motion into its horizontal and vertical components. The horizontal component of the velocity remains constant throughout the motion, while the vertical component changes due to the effect of gravity.
Given:
Height of the window (h) = 16.0 m
Launch angle with respect to the ground (θ) = 55.0°
Acceleration due to gravity (g) = 9.8 m/s²
Since the ball reaches the window, the vertical displacement (Δy) at the top of the projectile's trajectory will be equal to the height of the window.
Using the equation for vertical displacement:
Δy = (v₀ * sin(θ) * t) - (0.5 * g * t²)
Where:
v₀ = initial velocity of the projectile
sin(θ) = sine of the launch angle
t = time of flight
At the highest point, the vertical velocity (v_y) will be zero. We can determine the time of flight using the equation for vertical velocity:
v_y = v₀ * sin(θ) - g * t
Setting v_y equal to zero:
0 = v₀ * sin(θ) - g * t
Solving for t:
t = (v₀ * sin(θ)) / g
Now, we can substitute the value of t into the equation for vertical displacement:
h = (v₀ * sin(θ) * ((v₀ * sin(θ)) / g)) - (0.5 * g * ((v₀ * sin(θ)) / g)²)
Rearranging the equation:
(v₀ * sin(θ))² / 2g = h
Substituting the given values:
( (v₀ * sin(55.0°))² ) / (2 * 9.8) = 16.0
Simplifying the equation and solving for v₀:
v₀ = sqrt( (16.0 * 2 * 9.8) / sin²(55.0°) )
Now, we can calculate the value of v₀ using a calculator:
v₀ ≈ 27.3 m/s
Therefore, the boy should throw the ball with a velocity of approximately 27.3 m/s in order for it to reach his friend's window.