Question

You drop a set of keys as you walk down the street. Do the keys land behind you or right at your feet? Is the drop time of the keys affected by your forward motion? Is the acceleration of the keys affected by your forward motion? Explain.

Short answer

The keys land behind your initial position. Drop time and acceleration are unaffected by forward motion.

Step-by-step solution

Examining Key Landing Position

When you drop the keys while walking, they will tend to land behind your initial point of release. This happens because the keys retain the forward motion they had when released. However, to an observer moving alongside you, the keys appear to fall straight down at your feet due to the shared velocity.

Evaluating Drop Time Impact

The time it takes for the keys to hit the ground is guided purely by gravity, which is a vertically-acting force. Therefore, the horizontal motion of walking does not influence the time it takes for the keys to drop, meaning the drop time remains unaffected by your forward motion.

Analyzing Acceleration Effect

The acceleration of the keys in the drop process is affected solely by gravity, which acts downward at approximately 9.81 m/s². The forward motion of walking does not impact the acceleration, as motion is determined independently along each direction according to Newton's laws.

Key concepts

Gravity

Gravity is a fundamental force of nature that pulls objects toward the center of the Earth. It acts on everything with mass, giving weight to physical objects and causing them to fall to the ground when dropped. In the case of dropping keys, gravity is the primary force acting on them. When you let the keys go, gravity accelerates them downwards at a rate of approximately 9.81 m/s².

This acceleration is constant regardless of what happens in the horizontal direction. It means that whether you're standing still or moving forward, gravity's effect on the keys' fall remains unchanged.

Gravity ensures that when we drop something, it hits the ground in a predictable way, governed by consistent laws. It's crucial to understand that while gravity affects vertical motion, it doesn't influence horizontal movement.

Newton's Laws of Motion

Newton's Laws of Motion are three fundamental principles that describe the relationship between the motion of an object and the forces acting on it.

**First Law (Inertia)** states that an object will remain at rest, or in uniform motion in a straight line, unless acted upon by an external force. This explains why the keys retain their initial forward motion when released.

**Second Law (F=ma)** explains how the acceleration of an object depends on the net force acting upon it and its mass. In the keys' scenario, the force of gravity is what causes them to accelerate downwards.

**Third Law (Action and Reaction)** tells us that for every action, there is an equal and opposite reaction. When the keys fall, the Earth also exerts an opposite force, although the effect on the Earth's large mass is negligible.

These laws help us understand how objects move in different directions independently, which is why the keys' horizontal and vertical motions are separate.

Free Fall

Free fall occurs when an object moves under the influence of gravitational force alone, without resistance from other forces like air drag. In a vacuum, where there is no air resistance, all objects fall at the same rate regardless of their mass or shape.

In reality, air resistance can affect how different objects fall, but for small objects like keys in a short drop, this resistance is minimal.

When you drop your keys while walking, they are in a state of free fall in the vertical direction. This means they are accelerating straight down due to gravity alone, at a constant rate of 9.81 m/s².

Free fall is a crucial concept because it emphasizes the independence of vertical motion from any horizontal motion, such as the forward momentum of someone walking.

Velocity

Velocity describes both the speed and direction of an object's motion. It is a vector quantity, meaning it includes information about both how fast something is moving and where it is going.

When you walk and drop your keys, the keys inherit your forward velocity at the moment of release. Thus, they have two components of motion - forward and downward.

The forward velocity is what you initially impart to the keys, and it doesn't change because gravity acts vertically, not horizontally. Meanwhile, the velocity changes vertically due to gravitational acceleration.

• Horizontal velocity: Constant due to the inertia of the keys.
• Vertical velocity: Increases steadily due to gravity.

Understanding velocity is key to predicting where and how an object will move, which in this case, tells why the keys land slightly behind the release point but appear to fall straight down from our perspective.