To solve this problem, we can break it down into two separate motion problems: the horizontal motion and the vertical motion of the rock.
a) To determine the rock's time of flight, we can focus on the vertical motion. The rock is thrown with an initial velocity of 12 m/s at an angle of 25° above the horizon. The vertical component of the initial velocity can be calculated as follows:
Vertical component of initial velocity = 12 m/s * sin(25°) = 5.14 m/s
Since the rock is thrown vertically upwards and clears the surrounding moat, it will reach its highest point when it has zero velocity. From this highest point, it will then fall downwards. The time of flight is equal to the time it takes for the rock to reach its highest point and then fall back down.
To find the time it takes to reach the highest point, we can use the equation of motion:
Final velocity (v) = Initial velocity (u) + acceleration (a) * time (t)
Since the final velocity at the highest point is zero and the acceleration is due to gravity (g = 9.8 m/s^2), the equation becomes:
0 = 5.14 m/s - 9.8 m/s^2 * t
Solving for t, we get:
t = 0.524 seconds (rounded to 3 decimal places)
Therefore, the rock's time of flight is approximately 0.524 seconds.
b) To determine the width of the moat, we need to calculate the horizontal distance traveled by the rock. This can be done by focusing on the horizontal motion of the rock.
The horizontal component of the initial velocity can be calculated as follows:
Horizontal component of initial velocity = 12 m/s * cos(25°) = 10.87 m/s
The distance traveled by the rock can be calculated using the equation:
Distance (d) = Initial velocity (u) * time (t)
Substituting the known values, we get:
Distance (d) = 10.87 m/s * 0.524 s = 5.69 meters (rounded to 2 decimal places)
Therefore, the width of the moat is approximately 5.69 meters.
c) To determine the rock's velocity at impact, we can combine the horizontal and vertical components of velocity at the moment of impact.
At the highest point, the vertical component of velocity is zero. The horizontal component of velocity remains constant throughout the motion. Therefore, at the moment of impact, the rock's velocity will only have a horizontal component.
The horizontal component of velocity is:
Horizontal component of velocity = 10.87 m/s
Therefore, the rock's velocity at impact is approximately 10.87 m/s, in the horizontal direction.
Ex.4: A medieval prince locked in a castle attaches a note to a rock and throws it
at 12 m/s [25° above the horizon] from atop the castle's outer wall at a height
of 9.5 m so it just clears the surrounding moat. Determine: a) the rock's time of
flight, b) the width of the moat, c) the rock's velocity at impact.
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