Κάθετη κίνηση – προβλήματα και λύσεις
1. Ball A threw vertically upward with the ταχύτητα of 10 m/s. 1 second later, from the same position, Ball B is thrown vertically upward at the same path, with the speed of 25 m/s. What is the height of ball B when it encounters ball A.
Λύση:
In solving the problem of κατακόρυφη κίνηση, the vector quantity which direction upward is given a positive sign, the διάνυσμα quantity that direction downward is given a negative sign.
Γνωστό:
Αρχική ταχύτητα (vo) of ball A = 10 m/s
Time interval (t) of ball A = x
Αρχική ταχύτητα (vo) of ball B = 25 m/s
Time interval (t) of ball B = x – 1
Επιτάχυνση λόγω βαρύτητας (g) = -10 m/s2 (given minus sign because the direction of βαρύτητα is downward)
Ζητούνται: The height of ball B when it encounters ball A (h)
hA = hB
vo t + ½ gt2 = vo t + ½ gt2
10x + ½ (-10) x2 = 25 (x-1) + ½ (-10) (x-1)2
10x - 5x2 = 25 (x-1) – 5 (x-1)2
10x - 5x2 = 25x – 25 – 5 (x2-2x+1)
10x - 5x2 = 25x – 25 – 5x2 +10x –5
10x - 5x2 – 25x + 25 + 5x2 - 10x + 5 = 0
- 5x2 + 5χ2 + 10x – 25x – 10x + 25 + 5 = 0
10x – 25x – 10x + 25 + 5 = 0
– 25x + 25 + 5 = 0
- 25x + 30 = 0
– 25x = – 30
x = -30/-25
x = 1.2 seconds
Time interval ball A in air before it encounters ball B = 1.2 seconds
Time interval ball B in air before it encounters ball A = 1.2 seconds – 1 seconds = 0.2 seconds.
The height of ball A when it encounters ball B :
h = vo t + ½ gt2 = (10)(1.2) + 1/2 (-10)(1.2)2 = 12 – 5(1.44) = 12 – 7.2 = 4.8 μετρητήs
The height of ball B when it encounters ball A :
h = vo t + ½ gt2 = (25)(0.2) + 1/2 (-10)(0.2)2 = 5 – 5(0.04) = 5 – 0.2 = 4.8 μετρητήs
1. Ερώτηση: What is meant by vertical motion?
Απάντηση: Vertical motion refers to the movement of an object upward or downward, typically under the influence of gravitational force.
2. Ερώτηση: How is acceleration due to gravity (g) significant in vertical motion?
Απάντηση: All objects near Earth’s surface experience a constant acceleration, , which is approximately 9.81 m/s² downward.
3. Ερώτηση: Can an object have an initial velocity in vertical motion?
Απάντηση: Yes, objects can have an initial upward or downward velocity when their vertical motion starts.
4. Ερώτηση: What happens to the velocity of a freely falling object?
Απάντηση: The velocity of a freely falling object increases by approximately 9.81 m/s every second due to Earth’s gravity.
5. Ερώτηση: How is the time of ascent related to the time of descent for a vertically thrown object?
Απάντηση: For an object thrown upward and then allowed to fall back, the time of ascent equals the time of descent.
6. Ερώτηση: What is the velocity of an object at its maximum height?
Απάντηση: At maximum height, an object’s vertical velocity becomes zero before it starts descending.
8. Ερώτηση: How does air resistance affect vertical motion?
Απάντηση: Air resistance opposes motion, reducing acceleration and terminal velocity for falling objects.
9. Ερώτηση: What is terminal velocity?
Απάντηση: Terminal velocity is the constant maximum velocity reached by a falling object when air resistance equals the force of gravity.
10. Ερώτηση: Can an object have negative acceleration during upward motion?
Απάντηση: Yes, when an object moves upward against gravity, it has a negative acceleration equal to -g.
12. Ερώτηση: Why is the acceleration negative for objects thrown upward?
Απάντηση: Because acceleration due to gravity acts downward, it’s considered negative for objects moving in the opposite direction.
14. Ερώτηση: What is free fall?
Απάντηση: Free fall is the motion of an object under the sole influence of gravity, with no other forces acting on it.
15. Ερώτηση: How does the motion of an object differ when thrown downward versus when dropped?
Απάντηση: Both experience acceleration due to gravity. However, an object thrown downward has an additional initial velocity, making it reach the ground faster than one simply dropped.
16. Ερώτηση: What factors affect an object’s terminal velocity?
Απάντηση: Factors include object’s mass, shape, surface area, and the medium’s density and viscosity it’s falling through.
17. Ερώτηση: Does an object in vertical motion possess kinetic and potential energy?
Απάντηση: Yes, an object’s kinetic energy increases as it falls, while its potential energy decreases, and vice-versa during ascent.
18. Ερώτηση: Why does an object’s velocity change during vertical motion?
Απάντηση: The gravitational force causes a constant acceleration, changing the object’s velocity until it reaches terminal velocity or changes direction.
19. Ερώτηση: Can vertical motion be described as uniformly accelerated motion?
Απάντηση: Yes, in the absence of air resistance, vertical motion under gravity is uniformly accelerated with an acceleration of .
20. Ερώτηση: How is the conservation of energy principle applied to vertical motion?
Απάντηση: The sum of kinetic and potential energy remains constant during vertical motion, assuming no energy loss to air resistance.
Understanding vertical motion is essential in classical mechanics and has practical applications ranging from sports science to engineering and safety regulations.
