To find the velocity with which the projectile strikes the wall, we need to break down the motion into its horizontal and vertical components. You've already calculated the speed of projection, which is a great start! Now, let's analyze the situation step by step to determine the final velocity at the point of impact.
Understanding the Components of Motion
When a projectile is launched at an angle, its motion can be divided into two components: horizontal and vertical. The initial velocity can be resolved into these two components using trigonometric functions.
Initial Velocity Components
Given that the angle of projection is 60 degrees and the initial speed \( u \) is 21.8 m/s, we can find the horizontal (\( u_x \)) and vertical (\( u_y \)) components:
- Horizontal component: \( u_x = u \cdot \cos(60^\circ) = 21.8 \cdot 0.5 = 10.9 \, \text{m/s} \)
- Vertical component: \( u_y = u \cdot \sin(60^\circ) = 21.8 \cdot \frac{\sqrt{3}}{2} \approx 18.87 \, \text{m/s} \)
Time of Flight to the Wall
Next, we need to determine how long the projectile is in the air before it strikes the wall. Since the wall is 30 m away, we can use the horizontal motion to find the time \( t \):
Using the formula for horizontal distance:
Distance = Velocity × Time, we have:
30 m = 10.9 m/s × \( t \)
Solving for \( t \):
\( t = \frac{30}{10.9} \approx 2.75 \, \text{s} \)
Vertical Motion Analysis
Now that we have the time of flight, we can analyze the vertical motion to find the final vertical velocity (\( v_y \)) just before impact. The vertical motion is influenced by gravity, which accelerates the projectile downwards at approximately \( 9.81 \, \text{m/s}^2 \).
Using the equation for vertical motion:
\( v_y = u_y - g \cdot t \)
Substituting the values:
- Initial vertical velocity \( u_y \approx 18.87 \, \text{m/s} \)
- Acceleration due to gravity \( g \approx 9.81 \, \text{m/s}^2 \)
- Time \( t \approx 2.75 \, \text{s} \)
Calculating \( v_y \):
\( v_y = 18.87 - 9.81 \cdot 2.75 \approx 18.87 - 26.98 \approx -8.11 \, \text{m/s} \)
The negative sign indicates that the projectile is moving downwards when it strikes the wall.
Finding the Resultant Velocity
Now, we can find the magnitude of the resultant velocity (\( v \)) at the point of impact using the Pythagorean theorem:
\( v = \sqrt{u_x^2 + v_y^2} \)
Substituting the values:
- \( u_x = 10.9 \, \text{m/s} \)
- \( v_y \approx -8.11 \, \text{m/s} \)
Calculating \( v \):
\( v = \sqrt{(10.9)^2 + (-8.11)^2} \approx \sqrt{118.81 + 65.76} \approx \sqrt{184.57} \approx 13.6 \, \text{m/s} \)
Final Thoughts
The velocity with which the projectile strikes the wall is approximately 13.6 m/s. This value represents the speed at which the projectile impacts the wall, taking into account both the horizontal and vertical components of its motion. If you have any further questions or need clarification on any part of the process, feel free to ask!
