Certainly! Let's approach this problem step-by-step.
### Calculation of Velocity after 3 Seconds
1. Given data:
- Initial velocity ([tex]\(v_0\)[/tex]) = 100 m/s
- Angle of projection ([tex]\(\theta\)[/tex]) = 30 degrees
- Time ([tex]\(t\)[/tex]) = 3 seconds
- Acceleration due to gravity ([tex]\(g\)[/tex]) = 9.8 m/s²
2. Decomposition of the initial velocity:
- Horizontal component ([tex]\(v_{0x}\)[/tex]):
[tex]\[
v_{0x} = v_0 \cdot \cos(\theta)
\][/tex]
[tex]\[
v_{0x} = 100 \cdot \cos(30^\circ) = 100 \cdot \frac{\sqrt{3}}{2} \approx 86.60 \, \text{m/s}
\][/tex]
- Vertical component ([tex]\(v_{0y}\)[/tex]):
[tex]\[
v_{0y} = v_0 \cdot \sin(\theta)
\][/tex]
[tex]\[
v_{0y} = 100 \cdot \sin(30^\circ) = 100 \cdot \frac{1}{2} = 50 \, \text{m/s}
\][/tex]
3. Velocity components after 3 seconds:
- Horizontal velocity ([tex]\(v_x\)[/tex]) remains constant as there is no horizontal acceleration:
[tex]\[
v_x = v_{0x} = 86.60 \, \text{m/s}
\][/tex]
- Vertical velocity ([tex]\(v_y\)[/tex]) after time [tex]\(t\)[/tex]:
[tex]\[
v_y = v_{0y} - g \cdot t
\][/tex]
[tex]\[
v_y = 50 - 9.8 \cdot 3 = 50 - 29.4 = 20.6 \, \text{m/s} \, \text{(downward)}
\][/tex]
4. Magnitude of the velocity vector:
[tex]\[
v = \sqrt{v_x^2 + v_y^2}
\][/tex]
[tex]\[
v = \sqrt{(86.60)^2 + (20.6)^2} \approx \sqrt{7500 + 424.36} \approx \sqrt{7924.36} \approx 89 \, \text{m/s}
\][/tex]
### Calculation of Time When Acceleration is Perpendicular to Velocity
5. Understanding perpendicularity:
- Acceleration is due to gravity and acts vertically downwards.
- At time [tex]\(t\)[/tex], velocity will be perpendicular to acceleration if the vertical component of velocity is zero, as the horizontal component is still [tex]\(v_x\)[/tex].
6. Setting up the condition:
[tex]\[
v_y = v_{0y} - g \cdot t = 0
\][/tex]
[tex]\[
50 - 9.8 \cdot t = 0
\][/tex]
[tex]\[
t = \frac{50}{9.8} \approx 5.10 \, \text{seconds}
\][/tex]
### Summary of Results
- The velocity of the projectile after 3 seconds is approximately 89 m/s.
- The time when its acceleration is perpendicular to its velocity is approximately 5.10 seconds.
