Question

A car of mass \(500 \mathrm{~kg}\) is accelerating up an inclined plane 1 in 50 with an acceleration \(1 \mathrm{~m} / \mathrm{s}^{2}\). The power delivered by the engine at an instant is 600 Watts. The speed of the car at this instant is (neglect air resistance and rolling friction) (take \(\left.\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) (1) \(2 \mathrm{~m} / \mathrm{s}\) (2) \(1 \mathrm{~m} / \mathrm{s}\) (3) \(6 \mathrm{~m} / \mathrm{s}\) (4) \(10 \mathrm{~m} / \mathrm{s}\)

Short answer

Option (2): 1 m/s

Step-by-step solution

- Determine the angle of inclination

Given that the incline is '1 in 50,' this means the incline rises 1 meter vertically for every 50 meters horizontally. Therefore, the tangent of the incline angle \( \theta \) is \( \tan \theta = \frac{1}{50} \).

- Calculate component of gravitational force along the incline

The component of the gravitational force acting down the incline is \( mg \sin \theta \). Given \( m = 500 \text{ kg} \) and \( g = 10 \text{ m/s}^2 \), and approximating \( \sin \theta \approx \tan \theta \) for small angles, we get: \[ F_{gravity} = 500 \times 10 \times \frac{1}{50} = 100 \text{ N} \].

- Calculate the net force on the car

The car is accelerating at \( 1 \text{ m/s}^2 \). Using Newton's second law, \( F_{net} = ma \):\[ F_{net} = 500 \times 1 = 500 \text{ N} \].

- Determine the total force delivered by the engine

The total force delivered by the engine includes both overcoming gravity and providing acceleration: \[ F_{engine} = F_{gravity} + F_{net} = 100 \text{ N} + 500 \text{ N} = 600 \text{ N} \].

- Calculate the speed of the car

The power delivered by the engine at this instant is 600 Watts. Power \( P \) is given by the product of the force and the velocity of the car: \[ P = F_{engine} \times v \]. Solving for velocity \( v \): \[ 600 = 600 \times v \Rightarrow v = 1 \text{ m/s} \].

- Compare with the given options

The calculated speed of the car is \( 1 \text{ m/s} \). Therefore, the correct answer is option (2).

Key concepts

mechanics

Mechanics is a branch of physics dealing with the motion of objects and the forces acting on them. To solve the problem about the car on an inclined plane, we use several core principles of mechanics.
First, we establish the angle of inclination, compute gravitational forces along the slope, and apply Newton's second law.
Mechanics focuses on concepts like force, mass, acceleration, work, and energy. These concepts help explain how the car moves up the inclined plane.
Understanding mechanics is essential to analyze motion in various situations, not just inclined planes.

inclined plane

An inclined plane is a flat surface set at an angle, other than a right angle, against a horizontal surface. It helps reduce the force needed to raise objects by spreading that force over a longer distance.
When dealing with an inclined plane problem, the first step is calculating the angle of inclination. In this exercise, the incline was '1 in 50', meaning the plane rises 1 meter vertically for every 50 meters horizontally. This ratio shows the steepness of the incline and helps determine the gravitational force component along the incline.
Using simple trigonometry, we find that the tangent of the incline angle \( \theta \) is equal to \( \frac{1}{50} \). For small angles, we approximate \( \sin \theta \approx \tan \theta \). Thus, the gravitational force component can be calculated and used in further steps.

Newton's laws of motion

Newton's laws of motion are fundamental to understanding the movement of objects. The second law is particularly important in this exercise.
Newton's second law states that the net force acting on an object is equal to the mass of that object multiplied by its acceleration (\( F = ma \)).
Here, the car's mass (500 kg) and acceleration (1 m/s²) allow us to calculate the net force required to move the car up the incline. This net force includes overcoming gravitational pull on the incline (100 N as calculated) and providing the necessary acceleration (500 N).
This total force (600 N) determines the power required by the engine to maintain the car's velocity.

power and energy

Power is the rate at which work is done or energy is transferred. In this problem, the car engine provides power to move up the incline. The power delivered by the engine is given as 600 Watts.
Power is calculated using the formula: \( P = F \cdot v \), where \( F \) is the force (600 N) and \( v \) is the velocity. Rearranging gives: \( v = \frac{P}{F} = \frac{600}{600} = 1 \text{ m/s} \).
Thus, the speed of the car at that instant would be 1 m/s. Power also relates to energy since power is the rate of energy usage over time. In practical scenarios, understanding how power translates to the motion of vehicles like cars is crucial for efficiency and performance analysis.