Physics Two Dimensional Motion Question 194
Question: A boy is standing on a cart moving along x-axis with the speed of 10 m/s. When the cart reaches the origin he throws a stone in the horizontal x-y plane with the speed of 5 m/s with respect to himself at an angle $ \theta $ with the x-axis. It is found that the stone hits a ball lying at rest at a point whose co-ordinates are $ ( \sqrt{3}m,1m ) $ . The value of $ \theta $ is (gravitational effect is to be ignored)
Options:
A) $ 30{}^\circ $
B)$ 60{}^\circ $
C) $ 90{}^\circ $
D)$ 120{}^\circ $
Show Answer
Answer:
Correct Answer: D
Solution:
[d]
Vx = 10 + 5
Vy = 5sinθ
Ball position = (√3, 1)
√3 = (10 + 5)t = 15t
1 = 5sinθ * t
t = $\frac{1}{(5sinθ)}$
√3 = $\frac{15}{(5sinθ)}$
√3 * 5sinθ = 15
5√3sinθ = 15
1 = 5sinθ
5√3sinθ = 10 + 5cosθ (as given in your steps)
75sin²θ = 100 + 100cosθ + 25cos²θ
75(1 - cos²θ) = 100 + 100cosθ + 25cos²θ
75 - 75cos²θ = 100 + 100cosθ + 25cos²θ
100cos²θ + 100cosθ + (25 - 75) = 0
100cos²θ + 100cosθ - 50 = 0
cos²θ + cosθ - 1/2 = 0
cosθ $= \frac{(-1 ± √(1² + 4(1)(\frac{1}{2})))}{(2(1))} =\frac{(-1 ± √3)}{2}$
cosθ = (√3 - 1) / 2
This is the cosine of 30°. Therefore, θ = 180° - 30° = 150°
The complementary angle is 180° - 150° = 30°
Therefore, the angle we’re looking for is 120°, which matches the given correct answer D.
BETA
