A stone is thrown vertically upward with a speed of 24 m/s?

Since the stone is thrown upward, its acceleration is -9.8 m/s^2. Let’s use the following equation to determine its velocity at this height.

vf^2= vi^2 + 2 a d, a = -9.8

vf^2 = 24^2 + 2 -9.8 13

vf = ± √321.2

This is approximately ±17.9 m/s. Since the ball is rising, its velocity is positive. Let’s use the following equation to determine the time.

vf = vi – a * t

√321.2 = 24 – 9.8 * t

t = (24 – √321.2) ÷ 9.8

This is approximately 0.62 seconds. The next time the stone is at this height, its velocity is -√321.2.

Let’s use the same equation to determine the time.

-√321.2 = 24 – 9.8 * t

t = (√321.2 + 24) ÷ 9.8

This is approximately 4.28 seconds. To prove that this answer is correct, let’s use the following equation.

d = vi t + ½ a * t^2, a = -9.8

d = 24 (√321.2 + 24) ÷ 9.8 + ½ -9.8 * [(√321.2 + 24) ÷ 9.8]^2

This is exactly 13 meters. This proves that the time is correct. There are two answers because the velocity at this height is either +√321.2 m/s or -+√321.2 m/s. I hope this is helpful for you.

http://www.mathguide.com/lessons2/FlightProjectile...

If you go to the website above, you will see graphs of height versus time for a projectile. The graph is shaped like an inverted parabola. This is reason that the stone can have the same speed and height at two different times.

Solution:

(a) v² = u² + 2 a s

=> v² = 24² + 2 x (-9.8) x 13

=> v = 17.9 m/s

(b)

x = u t + 0.5 a t²

=> 13 = 24t + 0.5 x (-9.8) x t²

=> 4.9t² - 24t + 13 = 0 [<= quadratic equation]

=> t = 0.62 s or 4.28 s

time taken to reach the height of 13 m the first time round is 0.62 s.

(c)

From (b) above, we can see the quadratic equation can generate 2 answers.

However, that is not the reason for 2 answers to (b).

The reason is the stone has not reached its maximum height when it reaches the height of 13 m.

So, the stone reaches the height of 13 m first before it proceeds to its maximum height.

Thereafter, at time = 4.28 s, the stone falls back down to the height of 13 m the second time round.

Hence, there are two answers to (b).

*note: maximum height is 29.4 m

see calculation of maximum height:

K.E. at the lowest point = P.E. at the highest point

=> 0.5 m v² = m g h

=> v² = 2 g h

=> h = 0.5 (24)² / 9.8

= 29.4 m

hope this helps.