Problem 41 - Master Resource Book in JEE Main Physics

Question

A ball \(P\) is dropped vertically and another ball \(Q\) is thrown horizontally with the same velocities from the same height and at the same time. If air resistance is neglected, then (a) ball \(P\) reaches the ground first (b) ball \(Q\) reaches the ground first (c) both reach the ground at the same time (d) the respective masses of the two balls will decide the time

Step-by-Step Solution

Answer

(c) both reach the ground at the same time.

Step 1: Analyze the Motion of Ball P

Ball P is dropped vertically from a height with an initial vertical velocity of 0, since it is simply dropped and not thrown. The only force acting on ball P is gravity, causing it to accelerate downwards at 9.8 m/s². It will hit the ground when its initial height is reduced to zero.

Step 2: Analyze the Motion of Ball Q

Ball Q is thrown horizontally from the same height. Its horizontal motion does not affect how quickly it reaches the ground. Vertically, it behaves like a vertically dropped ball since the only force acting on it as well is gravity. Even if Q moves horizontally, its vertical acceleration is the same as ball P at 9.8 m/s².

Step 3: Compare Time to Hit the Ground

Since both balls started from the same height, are subject to the same gravitational acceleration, and have no vertical initial velocity, they will take the same amount of time to reach the ground regardless of their horizontal motion in the case of ball Q.

Key Concepts

Free Fall

Free fall is a concept in physics that describes the motion of objects under the influence of gravity alone. In the absence of other forces, such as air resistance, only gravity affects the falling object. This concept can simplify many physics problems, especially when considering vertical motion.

When an object is in free fall, it experiences a constant acceleration due to gravity. This gravitational acceleration is approximately 9.8 m/s² on Earth. This means that for every second an object is in free fall, its velocity increases by about 9.8 meters per second in the downward direction.

In the context of our exercise, both balls, P and Q, when considering only gravity, are in free fall. Despite any horizontal velocity ball Q might have, gravity pulls both balls downwards at the same rate. This is why in this exercise, the time taken for both balls to hit the ground is identical.

Vertical Motion

Vertical motion deals with the ascent or descent of an object in a straight line relative to gravity. The vertical component of projectile motion is influenced by gravity, which acts as a constant downward acceleration on the object. This concept is crucial for understanding how high or how quickly an object falls.

In the exercise, ball P is dropped and ball Q is thrown horizontally, but for both, the only vertical force acting is gravity. Ball P does not have any initial vertical velocity, while ball Q maintains a horizontal velocity that does not impact its vertical fall.
Here are key points to consider:

• Both balls are launched from the same height, affecting their vertical motion identically.
• The gravitational acceleration ensures their descent is identical in timing.
• Vertical motion of both balls is unaffected by any horizontal movements.

Ultimately, this means the rate at which both balls fall vertically is the same, leading them to strike the ground at the same time.

Horizontal Motion

Horizontal motion in physics describes how an object moves parallel to the ground. In projectile motion, an object’s horizontal motion is independent of its vertical motion. This concept helps to disentangle the horizontal travel from the vertical fall.

For ball Q, which is thrown horizontally, it retains a constant horizontal velocity because gravity does not impact horizontal motion. Ball P, being dropped without any horizontal component in its velocity, simply falls straight down.

Here are some insights into horizontal motion:

• Horizontal velocity does not affect how quickly an object falls to the ground.
• The horizontal and vertical motions are separate and do not influence each other.
• Other forces, like air resistance (neglected here), could alter horizontal motion.

Therefore, although ball Q travels horizontally while falling, its horizontal path does not alter the time it takes to reach the ground, ensuring it hits simultaneously with ball P.