Problem 44 - Master Resource Book in JEE Main Physics

Question

A ball is projected up an incline of \(30^{\circ}\) with a velocity of \(30 \mathrm{~ms}^{-1}\) at an angle of \(30^{\circ}\) with reference to the inclined plane from the bottom of the inclined plane. If \(g=10 \mathrm{~ms}^{-2}\), then the range on the inclined plane is (a) \(12 \mathrm{~m}\) (b) \(60 \mathrm{~m}\) (c) \(120 \mathrm{~m}\) (d) \(600 \mathrm{~m}\)

Step-by-Step Solution

Answer

None of the provided options match the calculation; re-examine conditions.

Step 1: Break down the velocity into components

Given the initial velocity \(v = 30\ \mathrm{ms}^{-1}\) and the angle \(\theta = 30^{\circ}\) with respect to the inclined plane, we find its components. The component along the incline can be found using \(v_{ ext{incline}} = v \cos(30^{\circ})\). Since the plane itself is inclined at \(30^{\circ}\), we need to adjust our calculations for the plane's inclination.

Step 2: Adjust velocities for incline

The effective component of velocity along the horizontal (base of the incline) is \(v_x = v \cos(\alpha + \beta)\), where \(\alpha = 30^{\circ}\) (velocity angle) and \(\beta = 30^{\circ}\) (incline angle). Thus, \(v_x = 30\ \cos(60^{\circ}) = 15\ \mathrm{ms}^{-1}\). The component along the incline is \(v_y = v \sin(\alpha + \beta) = 30\ \sin(60^{\circ}) = 25.98\ \mathrm{ms}^{-1}\).

Step 3: Determine the time of flight

The ball's motion up the inclined plane involves both horizontal and vertical components. To determine the time \(t\) for which the ball moves under gravity's influence, we use \(v_y = u_y - gt\) and resolve for \(t\). Here, \(u_y = v \sin(30^{\circ})\), \(g = g \cos(30^{\circ})\):\[ u_y = 30 \sin(30^{\circ}) = 15\ \mathrm{ms}^{-1} \]\[ g_{\text{effective}} = g \cos(30^{\circ}) = 10 \times \frac{\sqrt{3}}{2} = 8.66\ \mathrm{ms}^{-2} \]Thus, solving \(0 = 15 - 8.66t\) gives \(t = \frac{15}{8.66} \approx 1.73\ \mathrm{s}\).

Step 4: Calculate the range on the incline

The range \(R\) on the plane can be calculated using: \[ R = v_x \cdot t + \frac{1}{2} a_x \cdot t^2 \]Here, \(a_x = - g_{\text{effective}} \tan(30^{\circ})\) because it is the component of gravity acting on the incline:\[ a_x = - g \times \frac{1}{2} = -5 \mathrm{ms}^{-2} \]Thus,\[ R = 15 \times 1.73 + \frac{1}{2} \times (-5) \times (1.73)^2 = 25.95 - 7.48 = 18.47\ m\]

Step 5: Match result with given choices

Upon comparing our calculated inclination range of approximately \(18.47\ m\) with the given options, it seems incorrect. Recheck of the calculation shows no direct match suggesting further review. If assuming exact calculations have different conditions tied, values can result simplification errors which may still not align with direct options requiring practical boundaries.

Key Concepts

Inclined Plane Physics

In physics, an inclined plane is simply a flat surface tilted at an angle to the horizontal. This setup is often used to study the effects of forces and motion along surfaces that are not level. Imagine this incline as a triangle, where the tilt or angle is crucial for analyzing forces like gravity at work.

When dealing with projectile motion on an inclined plane, the movement of objects differs from horizontal motion. Here, the forces at play do not act uniformly because gravity pulls the object down the incline, affecting both its motion upwards and its eventual return down.

Studying inclined plane physics helps us understand how forces like friction or gravity affect movement differently on various planes, an important fundamental concept in physics.

Range of Projectile

The range of a projectile refers to how far it travels horizontally, or along any inclined plane, before coming to rest. Calculating this range on an inclined surface involves more than just a simple horizontal distance calculation, as the angle of inclination must be woven into the equations.

To determine the range on an inclined plane, we have to consider both the initial velocity of the projectile and the effects of gravity in the direction parallel to the incline. The effective gravitational pull can be adjusted using trigonometry to find how much it affects the motion along this plane.

• Initial velocity along the incline
• Time of flight determined by vertical motion equations
• Incline angle affecting effective gravity and acceleration

By understanding these components, we can see how the range is affected and apply this knowledge to other situations involving inclines.

Trigonometry in Physics

Trigonometry is a mathematical tool that plays a crucial role in understanding physics, especially when it comes to inclined planes and projectile motion. The use of trigonometric functions like sine, cosine, and tangent allows us to break down forces and velocities into components that are easier to analyze.

In the context of projectile motion on an inclined plane, trigonometry helps us:

• Calculate the components of velocity and acceleration parallel and perpendicular to the incline.
• Modify gravitational force effects using cosine for effective calculations.
• Apply the angle of projection to predict the motion path accurately.

Using these functions makes it simpler to understand complex motion and allows precise calculations for predicting projectile behavior under various scenarios.

Motion under Gravity

Motion under gravity is a fundamental concept in physics, describing how objects accelerate towards the Earth due to the gravitational force. When you're looking at projectile motion, especially on an inclined plane, understanding gravity's role is essential.

Gravity acts downward, but for inclined planes, this force is split into components.

• The component parallel to the incline affects how fast an object will slide or roll down.
• The perpendicular component presses the object against the plane, often affecting friction.

By recognizing these components, you can determine how an object will move along an inclined plane. In the case of the projectile exercise, we used gravity's influence to find both the effective acceleration along the incline and to modify the projectile's vertical motion, highlighting how gravity isn’t just about pulling things straight down. It's about how objects adjust to paths shaped by other forces and conditions.