Problem 64 - Master Resource Book in JEE Main Physics

Question

A particle \(A\) is projected from the ground with an initial velocity of \(10 \mathrm{~ms}^{-1}\) at an angle of \(60^{\circ}\) with horizontal. From what height \(h\) should an another particle \(B\) be projected horizontal with veloeity \(5 \mathrm{~ms}^{-1}\) go that both the particles collide with velocity \(5 \mathrm{~ms}^{-1}\) so that both the particles collide on the ground at point \(C\) if both are projected simultaneously? \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(10 \mathrm{~m}\) (b) \(30 \mathrm{~m}\) [c) \(15 \mathrm{~m}\) (d) \(25 \mathrm{~m}\)

Step-by-Step Solution

Answer

Check the document for steps.

Step 1: Identify given quantities

Extract initial velocity, acceleration, time, displacement.

Step 2: Choose kinematic equation

Use v=v0+at, x=v0t+0.5at^2, or v^2=v0^2+2a*dx.

Step 3: Substitute and solve

Plug in values and solve.

Step 4: State the answer

The answer is: Check the document for steps.

Key Concepts

Kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause the motion. To understand kinematics, it's important to remember that it involves the analysis of velocity, acceleration, time, and displacement in motion. These are fundamental quantities used to describe motion.

• **Velocity** refers to the speed of an object in a particular direction. For instance, the velocity vector of Particle A, in our problem, includes horizontal and vertical components, indicating how fast the particle moves in each direction.
• **Acceleration** is the rate of change of velocity over time. In the context of this problem, gravity (\(g = 10 ext{ m/s}^2\) acts as a downward constant acceleration on both particles.
• **Displacement** indicates the change in position of an object. In the context of the exercise, we focus on the entire path each particle takes until collision.

Kinematics equations are used to calculate various parameters, such as time of flight, range, and final velocity of the particles. A perfect understanding of kinematics is essential to solve problems related to projectile motion effectively.

Collision

When two particles meet at a point, a collision occurs. In physics, especially in our context of projectile motion, analyzing collisions involves determining the time and position where the two particles will meet.

• The collision point must be on the path trajectory of both particles, and it occurs at the same time for each.
• In this exercise, the two particles A and B collide at point C. This implies that they have the same horizontal position at the same instance.

To solve such a problem, it is necessary to calculate when both particles reach point C simultaneously, considering their velocities and trajectories. Understanding the behavior of materials after collision—whether it's elastic or perfectly inelastic—is deeper than what this problem requires, but it's a central idea in the study of collisions.

Horizontal Projectile

A horizontal projectile is an object that is launched horizontally from a height and allowed to move under the influence of gravity. In this problem, Particle B is an example of a horizontal projectile. It begins with an initial horizontal velocity and is acted upon by gravity.

• It has two distinct motions: a constant horizontal velocity and an accelerated vertical motion due to gravity.
• Unlike Particle A, which follows a curved trajectory, the horizontal motion of Particle B is uniform, meaning it moves in a straight line along the horizontal plane with constant velocity.

The initial height, from which the horizontal projectile is launched, significantly affects the time it takes to reach the ground. Therefore, calculating this initial height is crucial to predict where and when the projectile will hit the ground.

Angle of Projection

The angle of projection directly affects the trajectory of a projectile. Particle A in our example is launched at a specific angle of \(60^\circ\) relative to the horizontal. This angle influences several aspects of its motion:

• **Range:** The horizontal distance covered by the projectile. At different angles, the particle covers different ranges.
• **Maximum Height:** The highest vertical point in the trajectory. As the angle increases towards 90°, the maximum height increases.
• **Time of Flight:** The total time the object stays in motion. This time is influenced by both the angle and initial velocity.

The initial velocity should be broken into horizontal and vertical components using trigonometry. In this exercise, understanding the decomposed components helps to identify how high and how far the projectile will travel.