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1. A car moving too fast on a horizontal stretch of mountain road, slides off the road, falling into deep snow 50m below the road and 88m beyond the edge of this road.(a)How long did the car take to fall? (b) How fast was it going when it left the road?
Illustration: (include here the given)
II. Unknown:
Solution: (begin with your main equation/formula)
2. A stone is thrown horizontally at a speed of 5m/s from the top of a cliff 80m high. (a) How long does it take the stone to reach the bottom of the cliff? (b) How far from the base of the cliff does the stone strike the ground.? (c)What are the horizontal and vertical components of the velocity of the stone just before it hits the ground?
1. Illustration: (include here the given)
II. Unknown:
III. Solution: (begin with your main equation/formula)
1. A car moving too fast on a horizontal stretch of mountain road, slides off the road, falling into deep snow 50m below the road and 88m beyond the edge of this road.(a)How long did the car take to fall? (b) How fast was it going when it left the road?
Illustration: (include here the given)
II. Unknown:
Solution: (begin with your main equation/formula)
2. A stone is thrown horizontally at a speed of 5m/s from the top of a cliff 80m high. (a) How long does it take the stone to reach the bottom of the cliff? (b) How far from the base of the cliff does the stone strike the ground.? (c)What are the horizontal and vertical components of the velocity of the stone just before it hits the ground?
1. Illustration: (include here the given)
II. Unknown:
III. Solution: (begin with your main equation/formula)
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Answer:
1. a. 3.19 seconds to fall
b. 27.59 m/s (or 99.3 km/h) when it left the road.
2. a. 4.04 seconds for the stone to reach the bottom of the cliff.
b. 20.2 meters from the base of the cliff.
c. the horizontal component of the velocity is 5 m/s and the vertical component is approximately 39.6 m/s
Explanation:
1. We can solve this problem using the equations of motion for an object in free fall under the influence of gravity.
(a) To find how long the car took to fall, we can use the following equation of motion:
y = vi*t + 1/2*g*t^2
where y is the vertical distance fallen (50 m), vi is the initial vertical velocity (which we assume to be zero), g is the acceleration due to gravity (-9.81 m/s^2), and t is the time taken to fall.
Rearranging the equation and solving for t, we get:
t = sqrt(2*y/g)
t = sqrt(2*50/9.81)
t ≈ 3.19 seconds
Therefore, the car took approximately 3.19 seconds to fall.
(b) To find the initial horizontal velocity of the car when it left the road, we can use the following equation:
x = vi,h*t
where x is the horizontal distance traveled (88 m), vi,h is the initial horizontal velocity (what we want to find), and t is the time taken to fall (which we found in part (a)).
Rearranging the equation, we get:
vi,h = x/t
vi,h = 88/3.19
vi,h ≈ 27.59 m/s
Therefore, the car was going approximately 27.59 m/s (or 99.3 km/h) when it left the road.
2. (a) The time it takes for the stone to reach the bottom of the cliff can be found using the formula:
t = sqrt(2h/g)
where h is the height of the cliff (80 m) and g is the acceleration due to gravity (9.81 m/s^2).
Substituting the given values, we get:
t = sqrt(2*80/9.81) ≈ 4.04 seconds
Therefore, it takes approximately 4.04 seconds for the stone to reach the bottom of the cliff.
(b) The horizontal distance traveled by the stone can be found using the formula:
d = vt
where v is the horizontal velocity of the stone (which remains constant at 5 m/s throughout the motion) and t is the time taken to reach the bottom of the cliff (which we found in part (a)).
Substituting the given values, we get:
d = 5 * 4.04 ≈ 20.2 meters
Therefore, the stone strikes the ground approximately 20.2 meters from the base of the cliff.
(c) Just before hitting the ground, the stone has a horizontal velocity of 5 m/s and a vertical velocity of:
v = gt
where g is the acceleration due to gravity (9.81 m/s^2) and t is the time taken to reach the bottom of the cliff (which we found in part (a)).
Substituting the given values, we get:
v = 9.81 * 4.04 ≈ 39.6 m/s
Therefore, the horizontal component of the velocity is 5 m/s and the vertical component is approximately 39.6 m/s (upward, since the stone was thrown horizontally from the top of the cliff).