Problem 19 - College Physics

Question

Two identical stones are thrown from the top of a tall building. Stone 1 is thrown vertically downward with an initial speed \(v\), and stone 2 is thrown vertically upward with the same initial speed. Neglecting air resistance, which stone hits the ground with a greater speed: \((\) a) stone \(1,(\) b) stone \(2,\) or \((\mathrm{c})\) both have the same speed?

Step-by-Step Solution

Answer

(c) Both stones hit the ground with the same speed.

Step 1: Define Initial Conditions

Let's define that both stones are thrown from a height \( h \) above the ground. Stone 1 is thrown downward with an initial speed \( v \) and reaches its final speed as it hits the ground. Stone 2 is thrown upward with the same speed \( v \), rises to a maximum height before descending back past the throw point to hit the ground.

Step 2: Consider Stone 1's Motion

For stone 1, which is thrown downward:- Its initial speed \( v_{0_1} = v \).- Its acceleration \( a = g \) (where \(g\) is the acceleration due to gravity).- Apply the equation: \( v^2 = v_{0_1}^2 + 2gh \).Hence, the final speed of stone 1 when it hits the ground is \( v_1^2 = v^2 + 2gh \).

Step 3: Consider Stone 2's Motion

For stone 2, which is thrown upward:- Its initial speed \( v_{0_2} = v \), but in the upward direction so we consider \(-v\) when calculating the motion downwards.- At the highest point, its velocity becomes \(0\), then accelerates downward due to gravity.- From the highest point, apply the equation: \( v_2^2 = 0 + 2g(H + h) \) (where \( H \) is the additional height reached by stone 2). Since it falls back to \( h \), \( H = \frac{v^2}{2g} \), its speed then when falling back also becomes \(2gh + v^2 \) from the initial throw.- Simply put, the energy it loses going up is gained back going down, making \( v_2^2 = v^2 + 2gh \).

Step 4: Conclusion

After both calculations, we find that the final speed equations for both stones are identical: \( v_1^2 = v^2 + 2gh \) and \( v_2^2 = v^2 + 2gh \).This shows both stones hit the ground with the same final speed.

Key Concepts

Kinematics

Kinematics is the field of physics that describes the motion of objects without considering the forces that cause the motion. It focuses on parameters such as velocity, acceleration, displacement, and time. In this problem, we are examining two stones thrown in projectile motion. Each stone travels a different path, but kinematics helps us analyze their movements through space.

• Displacement: This is the distance and direction between an object's initial and final positions. For both stones, despite their different trajectories, the vertical displacement is the same as they both start from the same height and impacts the ground.
• Velocity: It describes the rate of change of position. Each stone has an initial velocity, which in this case is equal in magnitude but opposite in direction for stone 1 and stone 2.
• Acceleration: This refers to how quickly an object's velocity changes, usually due to gravity when objects are in free fall like the stones in our exercise.

Understanding these key concepts allows us to calculate how fast each stone will be moving when they hit the ground, revealing that both will have the same speed despite different initial directions.

Free Fall

Free fall occurs when an object moves only under the influence of gravity, without any other forces acting on it such as air resistance. In our exercise, we neglect air resistance, so both stones are essentially in free fall after they are launched, even though they initially experience different directions of motion.When an object is in free fall:

• The only force acting on it is gravity, which causes it to accelerate downwards.
• The acceleration due to gravity is a constant value, usually denoted by \( g \), which is approximately \( 9.81 \, \text{m/s}^2 \) near the surface of the Earth.
• No matter the initial upward or downward motion, once in free fall, velocity is dictated solely by gravity.

For stone 1, this means it will continually speed up as it descends downwards, directly in line with its initial direction. For stone 2, even though it first moves upward, it will eventually reverse direction due to gravity and fall downwards, accelerating just like stone 1 once it’s in free fall downward.

Acceleration Due to Gravity

Acceleration due to gravity is the rate at which an object's velocity changes as it moves under the influence of Earth's gravitational pull. It is a key factor in projectile motion and profoundly impacts the final velocities of both stones in our exercise.Key points about acceleration due to gravity:

• Constant Acceleration: Gravity provides a constant acceleration of approximately \( 9.81 \, \text{m/s}^2 \) vertically downwards. This constant force acts on all objects regardless of their initial motion.
• Impact on Motion:**: The acceleration affects both stones equally, regardless of their initial velocities, ensuring that both stones increase their velocity in the downward direction at the same rate.
• Independence of Direction:**: Whether stone 2 initially moves upward doesn’t affect the magnitude of gravity’s acceleration; it simply changes direction at its peak to follow the same acceleration path downwards as stone 1.

In conclusion, gravity acts equally on both stones after their initial throw regardless of their starting path, allowing us to deduce that they will arrive at the ground with the same speed.