Question

A stone is thrown vertically upward with a speed of 24.0 m/s. (a) How fast is it moving when it is at a height of 13.0 m? (b) How much time is required to reach this height? (c) Why are there two answers to (b)?

Answer 1

Answer:

Explanation:

initial speed, u = 24 m/s

acceleration due to gravity, g = - 9.8 m/s^2

(a) h = 13 m

Let it is moving with velocity v at a height of 13 m.

Use third equation of motion

$v^{2}=u^{2}+2gh$

By substituting the values

$v^{2}=24^{2}- 2 \times 9.8 \times 13$

v  = 17.9 m/s

(b) Let it takes time t to reach at height 13 m

Use second equation of motion

$s=ut+\frac{1}{2}gt^{2}$

$13=24t-4.9t^{2}$

$4.9t^{2}-24t+13=0$

$t=\frac{24\pm \sqrt{24^{2}-4\times4.9\times13}}{2\times 4.9}$

$t=\frac{24 \pm 17.92}{9.8$

t = 4.28 second or t = 0.62 second

(c) the value of t = 0.62 second shows that the journey is upward and at t = 4.28 second, the stone is in downward journey.

Answer 2

(a) The speed of the stone when it is 13 m upwards is 17.93 m/s

(b) The time required for the stone to reach the given height is 4.28 s or 0.62 s

(c) There are two values of time because the initial velocity is not zero and at the each of the given time, the height of the stone will be 13 m.

The given parameter:

• initial speed of the stone, u = 24 m/s

(a) The speed of the stone when it is 13 m upwards;

$v_y^2 = u^2 - 2gh\\\\v_y^2 = (24)^2 - (2\times 9.8 \times 13)\\\\v_y^2 = 321.2\\\\v_y = \sqrt{321.3} \\\\v_y = 17.9 3 \ m/s$

(b) The time required for the stone to reach the given height is calculated as;

$h= ut - \frac{1}{2} gt^2\\\\13 = 24t - (0.5\times 9.8)t^2\\\\13 = 24t - 4.9t^2\\\\4.9t^2 -24t + 13 = 0\\\\solve \ the \ quadratic \ equation \ as \ follows;\\\\a = 4.9, \ b = -24, \ c = 13\\\\t = \frac{-b \ \ +/- \ \ \sqrt{b^2 - 4ac} }{2a} \\\\t = \frac{-(-24) \ \ +/- \ \ \sqrt{(-24)^2 - (4\times 4.9 \times 13)} }{2(4.9)}$

$t = \frac{24 \ \+ /- \ \ 17.93}{9.8}\\\\t = \frac{24 \ + \ 17.93}{9.8} \ \ \ \ or \ \ \ \frac{24-17.93}{9.8} \\\\t = 4.28 \ s \ \ \ \ or \ \ \ \ 0.62 \ s$

(c) There are two values of time because the initial velocity is not zero and at the each of the given time, the height of the stone will be 13 m.

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