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Initial height of tower, h0 = 50 m

Mathematics

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3 parts
i

70 m\text{70 m}

ii

2 s\text{2 s}

iii

90 m\text{90 m}

Step-by-step solution

6 steps

Here are the calculations for the stone's motion:

Given:
Initial height of tower, h0=50 mh_0 = 50 \text{ m}
Initial velocity, u=20 m/su = 20 \text{ m/s} (upwards)
Acceleration due to gravity, g=10 m/s2g = 10 \text{ m/s}^2

Part i· 2 steps

The maximum height reached from the ground.

  1. Calculate the height gained by the stone above the tower.

    At maximum height, the final velocity v=0 m/sv = 0 \text{ m/s}. We use the kinematic equation v2=u2+2asv^2 = u^2 + 2as.
    Here, a=−ga = -g because gravity acts downwards, opposite to the initial upward motion.

    02=(20 m/s)2+2(−10 m/s2)s0^2 = (20 \text{ m/s})^2 + 2(-10 \text{ m/s}^2)s 0=400 m2/s2−20 m/s2⋅s0 = 400 \text{ m}^2/\text{s}^2 - 20 \text{ m/s}^2 \cdot s 20 m/s2⋅s=400 m2/s220 \text{ m/s}^2 \cdot s = 400 \text{ m}^2/\text{s}^2 s=400 m2/s220 m/s2=20 ms = \frac{400 \text{ m}^2/\text{s}^2}{20 \text{ m/s}^2} = 20 \text{ m}

    This is the height reached above the tower.

  2. Calculate the maximum height from the ground.

    The maximum height from the ground is the initial height of the tower plus the height gained above the tower.

    Hmax=h0+sH_{max} = h_0 + s Hmax=50 m+20 m=70 mH_{max} = 50 \text{ m} + 20 \text{ m} = 70 \text{ m}

    The maximum height reached from the ground is 70 m\boxed{\text{70 m}}.

Part ii

2 s\text{2 s}

The time it takes to reach the maximum height.

  1. Calculate the time to reach maximum height.

    We use the kinematic equation v=u+atv = u + at.
    Here, v=0 m/sv = 0 \text{ m/s}, u=20 m/su = 20 \text{ m/s}, and a=−g=−10 m/s2a = -g = -10 \text{ m/s}^2.

    0=20 m/s+(−10 m/s2)t0 = 20 \text{ m/s} + (-10 \text{ m/s}^2)t 10 m/s2⋅t=20 m/s10 \text{ m/s}^2 \cdot t = 20 \text{ m/s} t=20 m/s10 m/s2=2 st = \frac{20 \text{ m/s}}{10 \text{ m/s}^2} = 2 \text{ s}

    The time it takes to reach the maximum height is 2 s\boxed{\text{2 s}}.

Part iii

90 m\text{90 m}

The total distance covered.

  1. Calculate the distance covered during the upward motion.

    This is the height gained above the tower, which was calculated in part (i).
    Distance upward =20 m= 20 \text{ m}.

  2. Calculate the distance covered during the downward motion.

    The stone falls from its maximum height (70 m from the ground) to the ground.
    Distance downward =Hmax=70 m= H_{max} = 70 \text{ m}.

  3. Calculate the total distance covered.

    Total distance = Distance upward + Distance downward
    Total distance =20 m+70 m=90 m= 20 \text{ m} + 70 \text{ m} = 90 \text{ m}
    The total distance covered is 90 m\boxed{\text{90 m}}.

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