A stone is thrown vertically upward with a speed of 12.5 m/s from the edge of a cliff 65.0 m high (Fig. 2-34).
(a) How much later does it reach the bottom of the cliff?
(b) What is its speed just before hitting?
(c) What total distance did it travel?
(a) How much later does it reach the bottom of the cliff?
(b) What is its speed just before hitting?
(c) What total distance did it travel?
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The vertical motion Vy(t) = Vyi -g*t = 0 at the moment of peak altitude => Vyi/g = t
t = 12.5/9.81 = 1.27s
Use Vf^2 = Vi^2 + 2a*d to find the peak altitude when Vf = 0, a = -g and d = peak altitude
d = Vi^2/2g = 12.5^2/19.6 = 7.97m
To find the time to reach ground use d = do + Vi*t + 1/2 *a*t^2 where in this case do = 0, Vi = 0,a = g and
d = 65+7.97 = 72.97m
72.97 = 4.9*t^2 => t = sqrt[2*72.97/9.8] = 3.86s
Total time aloft is time to peak height plus time to ground
1.27+3.86 = 4.92s <----------------------------- (a)
Use V(t) = Vi + a*t = 0 + g*3.86 = 37.81m/s <----------------------------- (b)
Total distance traveled is up to peak altitude plus distance back to launch height plus 65m to ground
7.97+7.97+65 = 80.94m <----------------------------- (c)
t = 12.5/9.81 = 1.27s
Use Vf^2 = Vi^2 + 2a*d to find the peak altitude when Vf = 0, a = -g and d = peak altitude
d = Vi^2/2g = 12.5^2/19.6 = 7.97m
To find the time to reach ground use d = do + Vi*t + 1/2 *a*t^2 where in this case do = 0, Vi = 0,a = g and
d = 65+7.97 = 72.97m
72.97 = 4.9*t^2 => t = sqrt[2*72.97/9.8] = 3.86s
Total time aloft is time to peak height plus time to ground
1.27+3.86 = 4.92s <----------------------------- (a)
Use V(t) = Vi + a*t = 0 + g*3.86 = 37.81m/s <----------------------------- (b)
Total distance traveled is up to peak altitude plus distance back to launch height plus 65m to ground
7.97+7.97+65 = 80.94m <----------------------------- (c)
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A stone is thrown vertically upward with a speed of 12.5 m/s from the edge of a cliff 65.0 m high (Fig. 2-34).
(a) How much later does it reach the bottom of the cliff?
Distance = vi * t + ½ * g * t^2
65 = 12.5 * t + ½ * 9.8 * t^2
Solve this quadratic equation for time.
4.9 * t^2 + 12.5 * t – 65 = 0
t = [-12.5 ± (12.5^2 – 4 * 4.9 * -65)^0.5] ÷ 9.8
(12.5^2 – 4 * 4.9 * -65)^0.5 = 37.82
t = [-12.5 + 37.82] ÷ 9.8 = 2.58
t = [-12.5 – 37.82] ÷ 9.8 = -5.13 (neglect negative time)
(b) What is its speed just before hitting?
As the stone fell for 2.58 seconds, its velocity increased 9.8 m/s each second.
Final velocity = Initial velocity + 9.8 * t = 12.5 + 9.8 * 2.58 = 37.784 m/s
(c) What total distance did it travel?
The stone fell from the edge of a cliff 65.0 meters high to the ground, which is 65 meters below the edge of a cliff. So I believe the stone fell 65 meters!
Distance = 12.5 * 2.58 + ½ * 9.8 * 2.58^2 ≈ 64.9 m
The error is to rounding!
(a) How much later does it reach the bottom of the cliff?
Distance = vi * t + ½ * g * t^2
65 = 12.5 * t + ½ * 9.8 * t^2
Solve this quadratic equation for time.
4.9 * t^2 + 12.5 * t – 65 = 0
t = [-12.5 ± (12.5^2 – 4 * 4.9 * -65)^0.5] ÷ 9.8
(12.5^2 – 4 * 4.9 * -65)^0.5 = 37.82
t = [-12.5 + 37.82] ÷ 9.8 = 2.58
t = [-12.5 – 37.82] ÷ 9.8 = -5.13 (neglect negative time)
(b) What is its speed just before hitting?
As the stone fell for 2.58 seconds, its velocity increased 9.8 m/s each second.
Final velocity = Initial velocity + 9.8 * t = 12.5 + 9.8 * 2.58 = 37.784 m/s
(c) What total distance did it travel?
The stone fell from the edge of a cliff 65.0 meters high to the ground, which is 65 meters below the edge of a cliff. So I believe the stone fell 65 meters!
Distance = 12.5 * 2.58 + ½ * 9.8 * 2.58^2 ≈ 64.9 m
The error is to rounding!
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vo= 12.5m/s
a=gravitityconstant: g=9.8 m/s^2
height of the cliff: 65.0 m
use the formula: s(t)=-g t^2 /2 + vo t + 65.0
=-4.9 t^2 + 12.5 t +65.0
s(t)=0
0=-4.9 t^2 + 12.5 t +65.0
solve for t=5,13456
b) derivate s(t): v(t)= ds(t)/dt = -9.8 t + 12.5
and
v(5.13456)= -37.81
a=gravitityconstant: g=9.8 m/s^2
height of the cliff: 65.0 m
use the formula: s(t)=-g t^2 /2 + vo t + 65.0
=-4.9 t^2 + 12.5 t +65.0
s(t)=0
0=-4.9 t^2 + 12.5 t +65.0
solve for t=5,13456
b) derivate s(t): v(t)= ds(t)/dt = -9.8 t + 12.5
and
v(5.13456)= -37.81