Question

A ball is thrown straight up by a passenger in a train that is moving with a constant velocity. Where would the ball land-back in his hands, in front of him, or behind him? Does your answer change if the train is accelerating in the forward direction? If yes, how?

Short answer

Answer: The ball would land back in the passenger's hands when the train is moving with a constant velocity. In the scenario where the train is accelerating in the forward direction, the ball would land behind the passenger.

Step-by-step solution

Scenario 1: Train moving with constant velocity

When the train is moving with a constant velocity, both the train and the passenger share the same horizontal velocity. So, when the passenger throws the ball vertically, the ball also has the same horizontal velocity as the passenger and the train itself. Since the horizontal velocity does not change (it's constant), the ball will stay right above the passenger's hand during its entire vertical motion. Therefore, it would land back in his hands.

Scenario 2: Train accelerating in forward direction

When the train is accelerating, the situation is different. The passenger and the ball he throws still share the same initial horizontal velocity as the train. However, during the time the ball is in the air, the train's horizontal velocity continues to increase due to acceleration. Since the ball does not experience the train's acceleration (it's not in contact with the train after it's thrown), it's unable to keep up with the increasing horizontal velocity of the train. Consequently, the ball will land behind the passenger in this scenario. As the train accelerates even more, the difference between the train's position and where the ball lands becomes more significant.

Key concepts

Constant Velocity

Understanding the concept of constant velocity is important to predict the motion of an object when no external force is changing its speed or direction.
Imagine you're in a vehicle that's cruising steadily on a highway. Everything inside the vehicle is moving at the same speed and in the same direction as the vehicle itself. If you toss an object up into the air, it doesn't suddenly fly to the rear of the car. Why? Because the object maintains the vehicle's constant velocity, moving forward even while rising and falling.
In our train example, the ball that's thrown upward by the passenger would indeed land back in the passenger's hand. This outcome is because the ball's horizontal velocity matches that of the train and the passenger at the point of release. The vertical motion of the ball doesn't affect its horizontal motion, allowing it to fall back to its original spot, assuming no other forces act on it.

• The concept of inertia explains why an object in motion stays in motion with the same velocity unless acted upon by an external force.
• In a scenario with constant velocity, the horizontal motion and vertical motion are independent of each other.

Acceleration

Acceleration is a measure of how quickly an object changes its velocity. It can be a change in speed or direction or both. In physics, we describe acceleration as the rate at which the velocity of an object changes with time.
Returning to the train scenario, acceleration changes the game. The moment the ball leaves the passenger's hand, it has the same horizontal velocity as the train. However, since the train is accelerating, its velocity increases over time. The ball won't be affected by this change once thrown — it retains the velocity it had at the moment of release.
This means that while the ball is airborne, the train moves forward faster, and the ball falls behind. Hence, if the train is accelerating, the passenger will have to reach backward to catch the falling ball!

Key Points to Remember

• Acceleration involves a change in velocity over time.
• An object separated from an accelerating system (like the ball from the train) won't accelerate with it.
• The independence of vertical and horizontal motions is a key principle, but acceleration affects the horizontal position where the ball will land in relation to the train.

Projectile Motion

Projectile motion happens when an object is thrown into the air and moves along a curved path under the influence of gravity. This path is called a trajectory, and it's shaped like a parabola. The object is called a projectile, and its motion is called projectile motion.
In the train example, the ball is a projectile. The motion of the projectile is interesting because it happens in two dimensions: horizontal and vertical. These two aspects of motion are independent of one another.
Without external influences like air resistance, the horizontal component of the projectile's velocity remains constant, governed by the law of inertia. Meanwhile, the vertical component is influenced by gravity, leading to the characteristic 'arc' of projectile motion.

Interplay Between Vertical and Horizontal Motions

• The vertical motion is affected by gravity, causing the ball to rise and then fall back down.
• The horizontal motion continues at a constant velocity in the absence of other forces.
• In cases where the launch platform (like the train) is moving, the projectile retains the horizontal velocity of the platform at the moment of release.