You throw a ball vertically upward from the roof of a tall building. The ball leaves your hand at the point even with the roof railing with an upward speed of __15.0 m/s__, the ball is then in free fall. On its way back down, it just misses the railing. At the location of the building, g = __9.8m/s^2__.

Find :

**(a)** the *position* and the *velocity* of the ball __1.0s__ and __4.0s__ after leaving your hand.

**(b)** The *velocity* when the ball is __5.0m__ above the railing.

**(c) **The *maximum height* reached and the *time* at which it is reached.

**(d)** The *acceleration** *of the ball when it is at its maximum height.

**(e)** Find the *time* when the ball is __5.0m__ below the roof railing.

Guest Jul 31, 2017

#1**+1 **

**Hello Guest!**

You throw a ball vertically upward from the roof of a tall building. The ball leaves your hand at the point even with the roof railing with an upward speed of 15.0 m/s, the ball is then in free fall. On its way back down, it just misses the railing. At the location of the building, g = 9.8m/s^2.

Find :

(a) the position and the velocity of the ball 1.0s and 4.0s after leaving your hand.

(b) The velocity when the ball is 5.0m above the railing.

(c) The maximum height reached and the time at which it is reached.

(d) The acceleration of the ball when it is at its maximum height.

(e) Find the time when the ball is 5.0m below the roof railing.

**(a) the position and the velocity of the ball 1.0s and 4.0s after leaving your hand.**

\(h=vt-\frac{1}{2}gt^2\\ h_{1.0}=\frac{15m}{s}\cdot 1s-\frac{1}{2}\cdot\frac{9.8m}{s^2}\cdot 1^2s^2\\ \color{blue}h_{1.0}=10.1m\ over\ the\ railing\)

\(h_{4.0}=\frac{15m}{s}\cdot 4s-\frac{1}{2}\cdot\frac{9.8m}{s^2}\cdot 4^2s^2\\ h_{4.0}=60m-79.2m=-19.2m\\ \color{blue}h_{4.0}=19.2m\ under\ the\ railing\)

\(v=v_0-gt\\ v_{1.0}=\frac{15m}{s}-\frac{9.8m\cdot1s}{s^2}=5.2\frac{m}{s}\\ \color{blue}v_{1.0}=5.2\frac{m}{s}\ up \)

\(v_{4.0}=\frac{15m}{s}-\frac{9.8m\cdot4s}{s^2}=-24.2\frac{m}{s}\\ \color{blue}v_{4.0}=24.2\frac{m}{s}\ down\)

**(b) The velocity when the ball is 5.0m above the railing.**

\(v=v_0-\sqrt{2gh}\\ v_{(+5m)}=\frac{15m}{s}-\sqrt{\frac{2\cdot 9.8m\cdot 5m}{s^2}}=5.1\frac{m}{s}\\ \color{blue}v_{(+5m)}=5.1\frac{m}{s}\ up\)

**(c) The maximum height reached and the time at which it is reached. **

\(E=\frac{mv^2}{2}=mgh\\ h=\frac{mv^2}{2mg}=\frac{v^2}{2g}\)

\(h_{max}=\frac{15^2m^2s^2}{2\cdot 9.8ms^2}\\ \color{blue} h_{max}=11.480m\ over\ the\ railing. \)

\(v=gt\\ t_{rise}=\frac{v}{g}=\frac{15ms^2}{9.8ms}\\ \color{blue} t_{rise}=1.531s\)

**(d) The acceleration of the ball when it is at its maximum height.**

The ball is weightless throughout the journey.

For the ball, the acceleration is zero everywhere. It is not exposed to any force.

For the viewer the acceleration is at the highest point and in the case g = 9.8m / s²

**(e) Find the time when the ball is 5.0m below the roof railing.**

\(s=vt-\frac{1}{2}gt^2\\ \frac{1}{2}gt^2-vt+s=0\\ t^2-\frac{2v}{g}t+\frac{2s}{g}=0\)

\(t=-\frac{p}{2}+\sqrt{(\frac{p}{2})^2-q}\\ \color{blue} t=\frac{v}{g}+\sqrt{\frac{v^2}{g^2}-\frac{2s}{g}}\)

\(t=\frac{15m/s}{9.8m/s^2}+\sqrt{(\frac{15}{9.8}s)^2-\frac{-2\cdot 5m}{9.8m/s^2}}\\ t=1.531s+\sqrt{2.343s^2+1,020s^2}\\ t=1.531s+1.834s\\ \color{blue}t=3.365s\ (5m\ under\ the\ railing)\)

!

asinus
Jul 31, 2017

#2**0 **

Thanks asinus :)

Guest:

Are you supposed to use physics formulas or calculus ?

Do you understand asinus's answer and is that what you were after ?

Melody
Jul 31, 2017

#5**+3 **

Solutions for A - E

\(\text{Position and Time }\\ \begin{array}{|rcll|} \hline a_1) \text { position }\\ h &=& v_i(t) -\dfrac{1}{2} g(t)^2 \\ h &=& 15t - 4.9t^2 \\ t\tiny \; \text{ @} \small \text { 1 and 4}\\ h &=& 15(1) - 4.9(1)^2 &=&10.1m \\ h &=& 15(4) - 4.9(4)^2 &=&-18.4m \\ a_2) \text { velocity} \\ v_f &=& v_i – gt \\ v_f &=& 15 - 9.8t \\ \text{t @ 1 and 4} \\ v_f &=& 15 - 9.8(1) &=&5.20m/s \\ v_f &=& 15 - 9.8(4) &=& -24.2m/s \\ \hline \end{array}\\ \text { at 1 second: ball is 10.1m (above) and moving @ 5.20 m/s (up)} \\ \text { at 4 seconds: ball is -18.4m (below) and moving -24.2 m/s(down)} \\\)

\(b)\text {Gravity is slowing ball} \\ \begin{array}{|rcll|} \hline v^2 &=&v_0^2 +2(a)(\Delta y) \\ v^2 &=&(15)^2 +2(-9.8)(5m) \leftarrow \text{ use negative acceleration} \\ v &=&\pm \sqrt{127} \text{ (use positive result) } \\ v &=& 11.27m/s \\ \hline \end{array}\)

\(c)\text {Maximum height occurs when the vertical velocity is zero (0)} \\ \begin{array}{|rcll|} \hline a &=& \dfrac {(v_f – v_i)}{t} \\ t &=& \dfrac {(v_f – v_i)}{a} \\ t &=& \dfrac {(0 - 15)}{(- 9.8)} \\ t &=& \dfrac{-15}{-9.8} \\ t &=& 1.53s \\ \hline \end{array}\)

\(d) \text{ Acceleration is constant (g)} \\\)

\(e) \text{ Time after displacement }\\ \begin{array}{|rcll|} \hline t &=& \dfrac {(v_f – v_i)}{a} \\ v_f^2 &=& v_i^2 + 2(a)(\Delta y)\\ v_f &=& \pm \ sqrt{(v_i^2 + 2(a)(\Delta y)} \leftarrow \small \text {Use negative result; the ball is moving downward } \\ t &=& \dfrac{-\sqrt{(v_i^2 + 2(a)(\Delta y)} + (v_i)}{(-g)} \\ t &=& \dfrac{-\sqrt{(15^2 + 19.6*5)} + 15)}{-9.8}\\ t &=& 3.36s \\ \hline \end{array}\)

For related effects of gravity, see this.

GingerAle
Aug 1, 2017