Problem 55 - Master Resource Book in JEE Main Physics

Question

A body of mass \(m\) is thrown upward at an angle \(\theta\) with the horizontal with velocity \(v\). While rising up the velocity of the mass after \(t\) second will be (a) \(\sqrt{[v \cos \theta)^{2}+(v \cdot \sin \theta)^{2}}\) (b) \(\sqrt{(v \cos \theta-v \sin \theta)^{2}-g t}\) (c) \(\sqrt{v^{2}+g^{2} t^{2}-(2 v \sin \theta) g t}\) (d) \(\sqrt{v^{2}+g^{2} t^{2}-(2 v \cos \theta) g t}\)

Step-by-Step Solution

Answer

The correct answer is (c).

Step 1: Resolve Velocity Components

The initial velocity vector \(\vec{v}\) can be resolved into two perpendicular components: horizontal component \(v_x = v \cos \theta\) and vertical component \(v_y = v \sin \theta\). These components allow us to analyze the motion separately in horizontal and vertical directions.

Step 2: Analyze Horizontal Motion

The horizontal component of the velocity remains constant throughout the flight because there are no forces acting in the horizontal direction (assuming air resistance is negligible). Hence, the horizontal velocity at any time \(t\) is still \(v_x = v \cos \theta\).

Step 3: Analyze Vertical Motion

The vertical motion is influenced by gravity. Thus, the velocity \(v_y(t)\) at time \(t\) is given by the equation \(v_y(t) = v \sin \theta - g t\), where \(g\) is the acceleration due to gravity, acting downwards.

Step 4: Calculate Resultant Velocity

The resultant velocity \(v(t)\) at time \(t\) is the vector sum of the horizontal and vertical velocity components. This can be calculated using the Pythagorean theorem: \[ v(t) = \sqrt{(v \cos \theta)^2 + (v \sin \theta - g t)^2} \].

Step 5: Simplify the Expression

Simplifying \( v(t) = \sqrt{(v \cos \theta)^2 + (v \sin \theta - g t)^2} \) gives us \[ v(t) = \sqrt{v^2 \cos^2 \theta + (v \sin \theta)^2 - 2 v \sin \theta \cdot g t + (g t)^2} \], which can further be rearranged into: \[ v(t) = \sqrt{v^2 + g^2 t^2 - 2 v \sin \theta \cdot g t}\]. This expression matches option (c).

Key Concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause these movements. It focuses on parameters such as displacement, velocity, and acceleration. In this problem, we're dealing with projectile motion, a crucial aspect of kinematics.

Projectile motion is a type of two-dimensional motion that involves an object projected into the air, subjected only to the acceleration due to gravity. The motion can be split into two independent parts:

• Horizontal motion, where the velocity remains constant.
• Vertical motion, affected by gravitational acceleration.

These components help analyze projectile motion more straightforwardly. We break down complex movements into simpler, more predictable parts.

Vector Resolution

Vector Resolution involves breaking down a vector into two or more components. In projectile motion, we resolve the initial velocity vector into horizontal and vertical components. This technique simplifies the analysis of motion.

For instance, the initial velocity \((v)\) is divided into:

• Horizontal component: \(v_x = v \cos \theta\)
• Vertical component: \(v_y = v \sin \theta\)

These components allow us to treat the horizontal and vertical motions separately, making calculations more manageable. Vector resolution is fundamental in converting complex systems into simpler ones that are easier to solve mathematically.

Gravity

Gravity is the force of attraction between two bodies with mass. On Earth, gravity gives weight to physical objects and causes them to fall towards the ground when dropped. In kinematics, especially projectile motion, gravity significantly affects the vertical component of motion.

In this problem, the acceleration due to gravity is constant, represented by \(g\). It acts downward and influences the vertical velocity over time, often leading to equations like:

\(v_y(t) = v \sin\theta - g t\)

The term \(g t\) helps calculate how much the velocity decreases when the object moves against gravity. Gravity is crucial for determining how long an object remains airborne and its maximum height during motion.

Velocity Components

Velocity components refer to the parts of velocity in specified directions. For projectile motion, analyzing the horizontal and vertical components of velocity is essential.

Initially, the velocity is resolved into:

• Horizontal component \(v_x = v \cos \theta\), which remains constant unless affected by external forces.
• Vertical component \(v_y = v \sin \theta - g t\), which changes due to gravitational acceleration.

The resultant velocity at any time \(t\) is determined by combining these components through:

\(v(t) = \sqrt{(v \cos \theta)^2 + (v \sin \theta - g t)^2}\)

This equation gives a vector sum of the velocities in each direction. It is vital for understanding how fast and in which direction an object is moving at any point in time during its flight.