Answer :
Let's solve the problem where a stone is thrown vertically upward with an initial velocity of 15 meters per second.
### Step-by-Step Solution:
1. Identify the given values:
- Initial velocity ([tex]\( v_0 \)[/tex]) = 15 m/s
- Acceleration due to gravity ([tex]\( g \)[/tex]) = 9.8 m/s² (acting downwards)
2. Find the time taken to reach the highest point.
At the highest point, the final velocity ([tex]\( v \)[/tex]) is 0 m/s because the stone momentarily comes to rest before it starts to fall back down.
Using the equation of motion:
[tex]\[ v = v_0 - g \cdot t \][/tex]
Set [tex]\( v = 0 \)[/tex]:
[tex]\[ 0 = 15 - 9.8 \cdot t \][/tex]
Solve for [tex]\( t \)[/tex]:
[tex]\[ 9.8 \cdot t = 15 \implies t = \frac{15}{9.8} \approx 1.5306 \, \text{seconds} \][/tex]
3. Calculate the maximum height reached by the stone.
Use the kinematic equation for position:
[tex]\[ h = v_0 \cdot t - \frac{1}{2} g \cdot t^2 \][/tex]
Substitute the values:
[tex]\[ h = 15 \cdot 1.5306 - \frac{1}{2} \cdot 9.8 \cdot (1.5306)^2 \][/tex]
Simplify:
[tex]\[ h \approx 15 \cdot 1.5306 - \frac{1}{2} \cdot 9.8 \cdot 2.342 \][/tex]
[tex]\[ h \approx 22.959 - 11.47959 \approx 11.4796 \, \text{meters} \][/tex]
### Final Answer:
- The time it takes for the stone to reach the highest point is approximately [tex]\( 1.5306 \)[/tex] seconds.
- The maximum height reached by the stone is approximately [tex]\( 11.4796 \)[/tex] meters.
Therefore, the detailed step-by-step solution shows that the time to reach the highest point is 1.5306 seconds, and the maximum height attained by the stone is 11.4796 meters.
### Step-by-Step Solution:
1. Identify the given values:
- Initial velocity ([tex]\( v_0 \)[/tex]) = 15 m/s
- Acceleration due to gravity ([tex]\( g \)[/tex]) = 9.8 m/s² (acting downwards)
2. Find the time taken to reach the highest point.
At the highest point, the final velocity ([tex]\( v \)[/tex]) is 0 m/s because the stone momentarily comes to rest before it starts to fall back down.
Using the equation of motion:
[tex]\[ v = v_0 - g \cdot t \][/tex]
Set [tex]\( v = 0 \)[/tex]:
[tex]\[ 0 = 15 - 9.8 \cdot t \][/tex]
Solve for [tex]\( t \)[/tex]:
[tex]\[ 9.8 \cdot t = 15 \implies t = \frac{15}{9.8} \approx 1.5306 \, \text{seconds} \][/tex]
3. Calculate the maximum height reached by the stone.
Use the kinematic equation for position:
[tex]\[ h = v_0 \cdot t - \frac{1}{2} g \cdot t^2 \][/tex]
Substitute the values:
[tex]\[ h = 15 \cdot 1.5306 - \frac{1}{2} \cdot 9.8 \cdot (1.5306)^2 \][/tex]
Simplify:
[tex]\[ h \approx 15 \cdot 1.5306 - \frac{1}{2} \cdot 9.8 \cdot 2.342 \][/tex]
[tex]\[ h \approx 22.959 - 11.47959 \approx 11.4796 \, \text{meters} \][/tex]
### Final Answer:
- The time it takes for the stone to reach the highest point is approximately [tex]\( 1.5306 \)[/tex] seconds.
- The maximum height reached by the stone is approximately [tex]\( 11.4796 \)[/tex] meters.
Therefore, the detailed step-by-step solution shows that the time to reach the highest point is 1.5306 seconds, and the maximum height attained by the stone is 11.4796 meters.
