Question

Suppose the function \(s(t)\) represents the position (in feet) of a stone \(t\) seconds after it is thrown directly upward from 6 feet above Earth's surface. a. Is the stone's acceleration positive or negative when the stone is moving upward? Explain. b. What is the value of the instantancous velocity \(v(t)\) when the stone reaches its highest point? Explain.

Short answer

Answer: The stone's acceleration when moving upward is negative, and its instantaneous velocity at its highest point is 0.

Step-by-step solution

Determine the Sign of Acceleration

When the stone is moving upward, it is actually moving against the force of gravity. Gravity is a force that pulls objects downwards, towards the Earth's surface, which means that the acceleration due to gravity is negative. Therefore, the stone's acceleration when moving upward is negative.

Analyze the Highest Point

When the stone reaches its highest point, the stone is momentarily at rest. This is because it has stopped moving upward and is about to change direction to start moving downward due to Earth's gravity. At this point, the velocity of the stone changes from positive (moving upward) to negative (moving downward).

Instantaneous Velocity at the Highest Point

As mentioned in Step 2, the stone is momentarily at rest at its highest point. This means that the instantaneous velocity \(v(t)\) at this highest point is 0. This is because the stone's upward velocity has been reduced to zero by the force of gravity before changing direction and falling back down towards the Earth's surface. In summary: a. The stone's acceleration when moving upward is negative due to the force of gravity acting against the motion. b. The instantaneous velocity of the stone at its highest point is 0, as the stone is momentarily at rest before changing direction to move downward.

Key concepts

Acceleration

Acceleration is a fundamental concept in the physics of motion that describes how the velocity of an object changes over time. It is a vector quantity, which means it has both a magnitude and a direction.

• When an object speeds up, its acceleration is in the same direction as its motion.
• When it slows down, the acceleration is opposite to the direction of motion.

In the context of this exercise, the stone's acceleration is primarily determined by the force of gravity. This force always acts downward towards the center of the Earth, meaning that, when the stone is tossed upward, the acceleration is negative relative to its motion. Gravity causes a consistent downward acceleration of approximately \[ 9.8 \, \text{m/s}^2 \] or \[ 32 \, \text{ft/s}^2 \] in most practical scenarios on Earth. This results in the stone's upward motion slowing down until it eventually stops, ready to descend.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific point in time. Unlike average velocity, which is calculated over a period, instantaneous velocity provides a more pinpointed measure of motion. It's like looking at the speedometer in a car at a specific moment. For the stone in this scenario, when it reaches its highest point after being thrown upwards, its instantaneous velocity is exactly zero.

• At this peak, the stone has fully decelerated due to the negative acceleration (gravity) acting on it.
• Right after the highest point, the stone begins to accelerate in the opposite direction (downwards) as it descends.

Thus, the velocity at any particular fraction of time can tell you whether the object is still ascending, has reached its peak, or is beginning to fall. This concept is crucial for understanding motion trajectories.

Force of Gravity

The force of gravity is a natural phenomenon by which all things with mass are brought toward one another. Here on Earth, gravity gives weight to physical objects and is the force that pulls objects down towards the Earth's center.

• The gravitational force is responsible for the negative acceleration experienced by the stone when thrown upwards.
• Regardless of the motion's direction, gravity's force is always directed downwards.

Gravity is central to understanding why the stone slows as it moves up and accelerates as it comes down. It acts as a constant, unchanging force that affects all objects within Earth's environment equally, only dependent on their mass and distance from Earth's center. It demonstrates why, despite being thrown, the stone returns to the Earth rather than continuing into space.