Roof Path
A ball is thrown up onto a roof?
A ball is thrown up onto a roof, landing 3.10 s later at height h = 21.0 m above the release level. The ball’s path just before landing is angled at θ = 50.0° with the roof. (a) Find the horizontal distance d it travels. (Hint: One way is to reverse the motion, as if on videotape.) What are the (b) magnitude and (c) angle (relative to the horizontal) of the ball’s initial velocity?
I need help please. Anyone?
V = initial velocity of the ball
H = maximum height reached by the ball
t = time to reach max height
T = time the ball reaches the roof = 3.1
T – t = time it takes the ball to fall from max height to the roof
H – 21 = distance the ball falls from max height to the roof
g = acceleration of gravity = 9.81 m/s^2
Use the equations for calculating max height and for the ball to fall from max height to the roof.
H = (1/2)*g*t^2
H – 21 = (1/2*g*(T – t)^2 = (1/2*)g*(3.1 – t)^2
Put the first equation into the second for H.
(1/2)*g*t^2 – 21 = (1/2)*g*(3.1 – t)^2 = (1/2)*g*[9.61 - 6.2*t + t^2]
-21 = (1/2)*g*(9.61 – 6.2*t) = 4.905*(9.61 – 6.2*t)
-4.281 = 9.61 – 6.2*t
6.2*t = 13.891
t = 2.24 seconds
Now use this time to get the initial vertical velocity.
v = g*t = 9.81*2.24 = 21.98 m/s (22 m/s would be close enough)
Use the vertical impact velocity and angle to get the horizontal component of velocity at impact. This will equal the initial horizontal velocity since there are no forces acting in the horizontal.
Time to fall = 3.2 – 2.24 = 0.86 seconds
Vertical velocity = v = g*(time to fall) = 9.81*0.86 = 8.43 m/s
tan(50) = (vertical velocity)/(horizontal velocity)
Horizontal velocity = 8.43/tan(50) = 7.07 m/s
The initial velocity is then:
V = SQRT[(7.07)^2 + (21.98)^2]
V = 23.09 m/s
Angle is:
tan(Angle) = (vertical velocity)/(horizontal velocity) = 21.98/7.07
tan(Angle) = 3.1068
Angle = 1.259 radians = 72.16 degrees
The initial velocity is 23.09 m/s and the angle with the horizontal is 72.16 degrees.
Just noticed I forgot to get the distance. Distance is just the horizontal speed times the time of flight.
Distance = 7.07*3.1
Distance = 21.93 meters
So the ball traveled a horizontal distance of 21.93 meters
path of neo, on the roof top
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