Problem 46 - Master Resource Book in JEE Main Physics

Question

A piece of marble is projected from earth's surface with velocity of \(50 \mathrm{~ms}^{-1}, 2 \mathrm{~s}\) later it just clears a wall \(5 \mathrm{~m}\) high. What is the angle of projection? (a) \(45^{\circ}\) (b) \(30^{*}\) (c) \(60^{\circ}\) (d) None of these

Step-by-Step Solution

Answer

The angle of projection is approximately \(14.25^\circ\), so the answer is (d) None of these.

Step 1: Understand the Problem

The marble is projected with an initial velocity of \(50 \, \mathrm{ms}^{-1}\). After 2 seconds, it just clears a wall that is 5 m high. We are to determine the angle of projection.

Step 2: Use Vertical Motion Equation

The vertical motion can be described by the equation: \[ s = u_y t - \frac{1}{2} g t^2 \]where \(s = 5 \, \mathrm{m}\), \(u_y = u \sin \theta\), \(u = 50 \, \mathrm{ms}^{-1}\), \(g = 9.8 \, \mathrm{ms}^{-2}\), and \(t = 2 \, \mathrm{s}\).

Step 3: Substitute Values into Vertical Motion Equation

Substituting the known values into the equation:\[ 5 = (50 \sin \theta) \cdot 2 - \frac{1}{2} \cdot 9.8 \cdot (2)^2 \]Simplifying this leads to:\[ 5 = 100 \sin \theta - 19.6 \]\[ 24.6 = 100 \sin \theta \]

Step 4: Solve for \(\sin \theta\)

Rearrange the equation to solve for \(\sin \theta\):\[ \sin \theta = \frac{24.6}{100} = 0.246 \]

Step 5: Find Angle \(\theta\)

Use the inverse sine function to find \(\theta\):\[ \theta = \sin^{-1}(0.246) \approx 14.25^\circ \]As this doesn't match any of the given options precisely, the answer must be (d) None of these.

Key Concepts

Vertical Motion Equation

In projectile motion, the path of a projectile is influenced by its vertical motion. To understand this, we can use the vertical motion equation:\[s = u_y t - \frac{1}{2} g t^2\]Here, the vertical displacement \( s \) is determined by two factors:

• The initial vertical velocity \( u_y \), which is the vertical component of the initial velocity \( u \), calculated as \( u \sin \theta \); this shows how the angle influences motion.
• The influence of gravity \( g \), a constant at approximately 9.8 \( \mathrm{ms}^{-2} \), which decelerates the projectile's upward motion.

In this formula, \( t \) represents the time, which affects both the vertical rise due to the initial velocity and the fall due to gravity. It's critical to substitute these values carefully to predict the projectile's position at any given time. Understanding this formula allows you to see how angles and time interact under gravitational pull.

Angle of Projection

The angle of projection \( \theta \) is crucial in determining a projectile's path. It affects both horizontal and vertical components of motion.

• A steeper angle results in a greater vertical component, which means the object reaches a higher altitude but doesn't travel as far horizontally.
• A smaller angle increases the horizontal distance the projectile covers, but reduces its maximum height.

Using the vertical motion equation, we see how the angle \( \theta \) impacts the vertical displacement directly through \( u\sin \theta \). Even a slight change in the angle can significantly affect the projectile's trajectory.For instance, in our problem, an incorrect angle will either not clear the wall or overshoot, illustrating its critical importance in problems involving projectile motion. Thus, accurately determining \( \theta \) is pivotal when solving related physics problems.

Kinematics in Two Dimensions

Kinematics in two dimensions deals with motion where objects like projectiles move along curved paths. It combines linear kinematics in both horizontal and vertical planes.

• In horizontal motion, objects move at a constant velocity because gravity only affects vertical motion.
• In vertical motion, gravity influences the motion, creating a parabolic trajectory.

Understanding kinematics in two dimensions involves breaking the motion into these two separate components and analyzing them with respective principles and equations. This technique makes it easier to solve problems like the marble clearing a wall, as we examine how far up and how far across the marble travels over time. By separately analyzing both components, we can accurately predict the complete motion trajectory, ensuring that calculations are precise and account for how each dimension contributes to the overall movement.