Two stones are launched from the top of a tall building. One stone is thrown in a direction 30.0degree above the horizontal with a speed of 10.0m/s ; the other is thrown in a direction 30.0degree below the horizontal with the same speed. Which stone lands farther away from the building?
a. The stone thrown upward lands farther from the building.
b. The stone thrown downward lands farther from the building.
c. Both stones land the same distance from the building.
Mastering Physics : Part A
The stone thrown upwards spends more time in the air.
Part B :
The stone thrown upward lands farther from the building .
They both have the same horizontal velocity component, but the ball thrown upward will remain in the air longer.
So what does that tell you about the disatnce that they travel? Think about it.
To determine which stone lands farther away from the building, we can calculate the horizontal distances traveled by each stone.
First, let's calculate the horizontal distance traveled by the stone thrown upward.
1. We can break down the initial velocity of the upward stone into horizontal and vertical components.
The horizontal component can be calculated as follows:
Horizontal component = Initial velocity * cos(angle)
= 10.0 m/s * cos(30.0°)
= 10.0 m/s * 0.866
= 8.66 m/s
2. The time it takes for the stone to reach the ground can be obtained using the vertical component of the initial velocity and considering acceleration due to gravity.
The vertical component can be calculated as follows:
Vertical component = Initial velocity * sin(angle)
= 10.0 m/s * sin(30.0°)
= 10.0 m/s * 0.5
= 5.0 m/s
The time taken for the stone to reach the ground can be calculated using the equation:
Vertical distance = (1/2) * acceleration * time^2
0 (because the stone falls to the ground) = (1/2) * (-9.8 m/s^2) * time^2
time^2 = 0
time = 0 s
Since the time taken is 0 s, the stone thrown upward never reaches the ground, so its horizontal distance is also 0.
Next, let's calculate the horizontal distance traveled by the stone thrown downward.
1. We can again break down the initial velocity of the downward stone into horizontal and vertical components.
The horizontal component can be calculated as follows (same as the upward stone):
Horizontal component = Initial velocity * cos(angle)
= 10.0 m/s * cos(30.0°)
= 8.66 m/s
2. The time it takes for the stone to reach the ground can be determined using the vertical component of the initial velocity, considering acceleration due to gravity.
The vertical component can be calculated as follows:
Vertical component = Initial velocity * sin(angle)
= 10.0 m/s * sin(30.0°)
= 10.0 m/s * (-0.5)
= -5.0 m/s
The time taken can be calculated as:
Vertical distance = (1/2) * acceleration * time^2
0 = (1/2) * (-9.8 m/s^2) * time^2
time^2 = 0
time = 0 s
Since the time taken is also 0 s, the stone thrown downward also does not hit the ground, so its horizontal distance is also 0.
Therefore, both stones land the same distance from the building, so the answer is option c. Both stones land the same distance from the building.
To find out which stone lands farther away from the building, we can analyze the horizontal components of their motion. Let's break down the problem step by step:
1. Calculate the horizontal component of the velocity for each stone:
The horizontal component of the velocity is given by Vx = V * cos(θ), where V is the initial speed and θ is the angle of projection.
For the stone thrown upward:
Vx1 = 10.0 m/s * cos(30.0°)
For the stone thrown downward:
Vx2 = 10.0 m/s * cos(-30.0°) (since the angle is below the horizontal, it will be negative)
2. As the stones are thrown from the same height and the time of flight is the same, the distance traveled horizontally by each stone is directly proportional to their horizontal velocities.
3. Compare the magnitudes of Vx1 and Vx2 to determine which is greater. The stone with the larger horizontal velocity will travel farther.
If Vx1 > Vx2, then the stone thrown upward lands farther from the building.
If Vx1 < Vx2, then the stone thrown downward lands farther from the building.
If Vx1 = Vx2, then both stones land the same distance from the building.
Therefore, to determine the correct answer, calculate the horizontal velocities Vx1 and Vx2 and compare their magnitudes.