Answer:
To determine the range of the marble when fired horizontally, we need to consider the horizontal motion of the marble. Since the marble is fired horizontally, its initial vertical velocity is zero, and it will only experience the horizontal motion due to gravity.
We can use the equation for horizontal motion:
Range = Horizontal Velocity × Time
To find the horizontal velocity, we need to determine the time it takes for the marble to reach the ground when fired horizontally. We can use the equation for vertical motion to find the time of flight:
h = (1/2) × g × t^2
where:
h = vertical displacement (6.0 m)
g = acceleration due to gravity (9.8 m/s^2)
t = time of flight
Rearranging the equation, we get:
t = sqrt((2h) / g)
Substituting the given values:
t = sqrt((2 × 6.0 m) / 9.8 m/s^2)
t ≈ 1.22 s
Now, we can calculate the range:
Range = Horizontal Velocity × Time
The horizontal velocity remains constant throughout the motion, so we can use the equation:
Range = Velocity × Time
The vertical displacement has no effect on the horizontal motion, so the time of flight is the same as when the marble was fired vertically.
Substituting the given values:
Time = 1.22 s
Range = Velocity × 1.22 s
However, we need to find the horizontal velocity. Since the marble is fired horizontally, it doesn't have any initial vertical velocity. Therefore, the horizontal velocity will be the same as the horizontal component of the initial velocity when it was fired vertically.
To find the initial vertical velocity when fired vertically, we can use the equation for vertical motion:
v = u + gt
Where:
v = final vertical velocity (0 m/s)
u = initial vertical velocity
Rearranging the equation, we get:
u = -gt
Substituting the given values:
g = 9.8 m/s^2
t = 1.22 s
u = -(9.8 m/s^2) × (1.22 s)
u ≈ -11.96 m/s
The negative sign indicates that the initial vertical velocity is directed downward.
Now, we can use the initial vertical velocity to find the horizontal velocity:
Horizontal Velocity = initial vertical velocity
Horizontal Velocity ≈ -11.96 m/s
Finally, we can calculate the range:
Range = Horizontal Velocity × Time
Range = (-11.96 m/s) × (1.22 s)
Range ≈ -14.59 m
The negative sign indicates that the range is in the opposite direction of the initial horizontal velocity. Therefore, the marble's range, when fired horizontally from 2.0 m above the ground, is approximately 14.59 meters in the opposite direction.