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Step-by-Step Solution
Step 1: Identify the Relevant Physical Quantities
A projectile (the marble) is thrown with an initial speed of
50\,\text{m/s} from the ground. After 2\,\text{s} of flight, it just clears a wall of height 5\,\text{m} . Let v_0 be the initial velocity, \theta the angle of projection, g the acceleration due to gravity ( 9.8\,\text{m/s}^2 ), and t the time elapsed (which here is 2\,\text{s} ).
Step 2: Establish the Vertical Motion Equation
The vertical position y of the projectile at time t is given by the standard equation for vertical displacement:
y = (v_0 \sin \theta)\,t \;-\; \tfrac{1}{2} g\,t^2
In this problem, after t = 2\,\text{s} , the marble just clears a 5\,\text{m} high wall. Therefore, y = 5\,\text{m} when t=2\,\text{s} .
Step 3: Substitute the Known Values
Substitute v_0 = 50\,\text{m/s} , t = 2\,\text{s} , and y = 5\,\text{m} into the equation:
5 = \bigl(50 \sin \theta\bigr) \times 2 \;-\; \tfrac{1}{2}\times 9.8 \times (2)^2
Step 4: Simplify the Equation
First compute the second term on the right-hand side:
\tfrac{1}{2} \times 9.8 \times (2)^2 = \tfrac{1}{2} \times 9.8 \times 4 = 19.6
So the equation becomes:
5 = 100 \sin \theta - 19.6
Rearrange to isolate \sin \theta :
100 \sin \theta = 5 + 19.6 = 24.6
\sin \theta = \tfrac{24.6}{100} = 0.246
Step 5: Calculate 4 \sin \theta
We need 4 \sin \theta :
4 \sin \theta = 4 \times 0.246 = 0.984
Final Answer
The value of 4 \sin \theta is approximately 0.984 .
