3. (a) In a room of height 6 m, a ball is projected from a point P. P is 1.1 m above the floor. The velocity of projection is 9.8√2 ms⁻¹ at an angle of 45° to the horizontal. The ball strikes the ceiling at Q without first striking a wall. Find the length of the straight line PQ. (b) A particle is projected up an inclined plane with initial speed 80 m s⁻¹. The line of projection makes an angle of 30° with the inclined plane and the plane is inclined at an angle θ to the horizontal. The plane of projection is vertical and contains the line of greatest slope. The particle strikes the plane at an angle of tan⁻¹(−2/√3). Find (i) the value of θ (ii) the speed with which the particle strikes the plane.

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3. (a) In a room of height 6 m, a ball is projected from a point P - Leaving Cert Applied Maths - Question 3 - 2010

Question 3

3. (a) In a room of height 6 m, a ball is projected from a point P. P is 1.1 m above the floor. The velocity of projection is 9.8√2 ms⁻¹ at an angle of 45° to the... show full transcript

Worked Solution & Example Answer:3. (a) In a room of height 6 m, a ball is projected from a point P - Leaving Cert Applied Maths - Question 3 - 2010

Step 1

(a) Find the length of the straight line PQ.

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Answer

To find the length of PQ, we need to determine the coordinates of points P and Q.

Identify Coordinates:
- Point P is at (0, 1.1) m (0 m horizontal position, 1.1 m above the floor).
- The height of the room is 6 m, making the maximum height of Q equal to 6 m.
Ball Projections:
- The horizontal and vertical components of the initial velocity (v₀) are:
  
  $v_{0x} = v_0 imes ext{cos}(45°) = 9.8√2 imes rac{1}{ ext{√2}} = 9.8 ext{ m s}^{-1}$
  
  $v_{0y} = v_0 imes ext{sin}(45°) = 9.8√2 imes rac{1}{ ext{√2}} = 9.8 ext{ m s}^{-1}$
Use Kinematics to Find Time to Reach Ceiling:
- Using the equation for vertical motion:
  
  $s = v_{0y}t - rac{1}{2}gt^2$
  
  Setting s = 6 m - 1.1 m = 4.9 m:
  
  $4.9 = 9.8t - rac{1}{2}(9.8)t^2$
Solve for t:
- Rearranging gives:
  
  $4.9 = 9.8t - 4.9t^2$
  
  Using the quadratic formula or simplifying, we find:
  
  $t ≈ 1 ext{ (considering only the positive root)}$
Calculate Horizontal Distance (x):
- Substitute t back into the horizontal distance:
  
  $x = v_{0x} imes t = 9.8 imes 1 ≈ 9.8 ext{ m}$
Find Length of PQ:
- The length PQ is given by:
  
  $|PQ| = ext{hypotenuse} = ext{√}(4.9^2 + 9.8^2)$
  
  $= √(24.01 + 96.04) ≈ 10.96 ext{ m}$

Step 2

(b) Find (i) the value of θ.

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Answer

Given Information:
- Initial speed, u = 80 m s⁻¹.
- Angle of projection with the incline = 30°.
- Resolve the initial velocity into components parallel and perpendicular to the incline.
Component Calculation:
- The y-component of velocity at the moment of striking:
  
  $v_{y} = u imes ext{sin}(30°) - g imes t$
Use Condition for Striking Plane:
- As the particle strikes the plane, we have:
  
  $r_f = 0$ and $0 = 80 imes ext{sin}(30°) - rac{1}{2}gt^2$
Calculate using Substitutions:
- Setting up equations using the above conditions leads to solving for θ such that:
  
  $tan θ = rac{-v_y}{v_x} = rac{40 - 80}{80 ext{cos}(30°)}$
  
  After simplifying, we find:
  
  $θ ≈ 23.4°$

Step 3

(b) Find (ii) the speed with which the particle strikes the plane.

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Answer

Calculate Components at Impact:
- The x-component at strike:
  
  $v_{x} = u imes ext{cos}(30°)$
- Substituting values gives:
  
  $v_{x} = 80 imes rac{ ext{√3}}{2} = 40 ext{√3} m/s$
Calculate y-component at Impact:
- Using previous y-component relationship, we have:
  
  $v_{y} = -40 ext{√3} ext{ (downward)}$
Finding the Resultant Speed:
- Total speed upon striking the plane:
  
  $ext{speed} = ext{√}(v_{x}^2 + v_{y}^2)$
  
  $= ext{√}((40 ext{√3})^2 + (-40)^2)$
  
  $= ext{√}(4800 + 1600) = ext{√6400} = 80 ext{ m s}^{-1}$

3. (a) In a room of height 6 m, a ball is projected from a point P - Leaving Cert Applied Maths - Question 3 - 2010

Worked Solution & Example Answer:3. (a) In a room of height 6 m, a ball is projected from a point P - Leaving Cert Applied Maths - Question 3 - 2010

(a) Find the length of the straight line PQ.

(b) Find (i) the value of θ.

(b) Find (ii) the speed with which the particle strikes the plane.

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