Vertical motion – problems and solutions

1. Ball A threw vertically upward with the speed 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.

Solution :

In solving the problem of vertical motion, the vector quantity which direction upward is given a positive sign, the vector quantity that direction downward is given a negative sign.

__Known :__

Initial speed (v_{o}) of ball A = 10 m/s

Time interval (t) of ball A = x

Initial speed (v_{o}) of ball B = 25 m/s

Time interval (t) of ball B = x – 1

Acceleration due to gravity (g) = -10 m/s^{2 }(given minus sign because the direction of gravity is downward)

__Wanted :__ The height of ball B when it encounters ball A (h)

h_{A} = h_{B}

v_{o} t + ½ g t^{2 }= v_{o} t + ½ g t^{2}

10x + ½ (-10) x^{2 }= 25 (x-1) + ½ (-10) (x-1)^{2}

10x – 5x^{2 }= 25 (x-1) – 5 (x-1)^{2}

10x – 5x^{2 }= 25x – 25 – 5 (x^{2}-2x+1)

10x – 5x^{2 }= 25x – 25 – 5x^{2 }+ 10x – 5

10x – **5x**^{2} – 25x + 25 + **5x**^{2} – 10x + 5 = 0

– 5x^{2} + 5x^{2 }+ 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 = v_{o} t + ½ g t^{2 }= (10)(1.2) + 1/2 (-10)(1.2)^{2 }= 12 – 5(1.44) = 12 – 7.2 = **4.8 meter****s**

The height of ball B when it encounters ball A :

h = v_{o} t + ½ g t^{2 }= (25)(0.2) + 1/2 (-10)(0.2)^{2 }= 5 – 5(0.04) = 5 – 0.2 = **4.8 meter****s**

**1. Question:** What is meant by vertical motion?

**Answer:** Vertical motion refers to the movement of an object upward or downward, typically under the influence of gravitational force.

**2. Question:** How is acceleration due to gravity (g) significant in vertical motion?

**Answer:** All objects near Earth’s surface experience a constant acceleration, $g$, which is approximately 9.81 m/s² downward.

**3. Question:** Can an object have an initial velocity in vertical motion?

**Answer:** Yes, objects can have an initial upward or downward velocity when their vertical motion starts.

**4. Question:** What happens to the velocity of a freely falling object?

**Answer:** The velocity of a freely falling object increases by approximately 9.81 m/s every second due to Earth’s gravity.

**5. Question:** How is the time of ascent related to the time of descent for a vertically thrown object?

**Answer:** For an object thrown upward and then allowed to fall back, the time of ascent equals the time of descent.

**6. Question:** What is the velocity of an object at its maximum height?

**Answer:** At maximum height, an object’s vertical velocity becomes zero before it starts descending.

**8. Question:** How does air resistance affect vertical motion?

**Answer:** Air resistance opposes motion, reducing acceleration and terminal velocity for falling objects.

**9. Question:** What is terminal velocity?

**Answer:** Terminal velocity is the constant maximum velocity reached by a falling object when air resistance equals the force of gravity.

**10. Question:** Can an object have negative acceleration during upward motion?

**Answer:** Yes, when an object moves upward against gravity, it has a negative acceleration equal to -g.

**12. Question:** Why is the acceleration negative for objects thrown upward?

**Answer:** Because acceleration due to gravity acts downward, it’s considered negative for objects moving in the opposite direction.

**14. Question:** What is free fall?

**Answer:** Free fall is the motion of an object under the sole influence of gravity, with no other forces acting on it.

**15. Question:** How does the motion of an object differ when thrown downward versus when dropped?

**Answer:** 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. Question:** What factors affect an object’s terminal velocity?

**Answer:** Factors include object’s mass, shape, surface area, and the medium’s density and viscosity it’s falling through.

**17. Question:** Does an object in vertical motion possess kinetic and potential energy?

**Answer:** Yes, an object’s kinetic energy increases as it falls, while its potential energy decreases, and vice-versa during ascent.

**18. Question:** Why does an object’s velocity change during vertical motion?

**Answer:** The gravitational force causes a constant acceleration, changing the object’s velocity until it reaches terminal velocity or changes direction.

**19. Question:** Can vertical motion be described as uniformly accelerated motion?

**Answer:** Yes, in the absence of air resistance, vertical motion under gravity is uniformly accelerated with an acceleration of $g$.

**20. Question:** How is the conservation of energy principle applied to vertical motion?

**Answer:** 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.