Problem 1
Question
An automobile travelling with a speed of \(60 \mathrm{kmh}\) can brake to stop with a distance of \(20 \mathrm{~m}\). If the car is going twice as fast i.e., \(120 \mathrm{kms}^{-1}\), the stopping distance will be (a) \(20 \mathrm{~m}\) (b) \(40 \mathrm{~m}\) (c) \(60 \mathrm{~m}\) (d) \(80 \mathrm{~m}\)
Step-by-Step Solution
Verified Answer
The stopping distance is 80 m, choice (d).
1Step 1: Understand the Relationship Between Speed and Stopping Distance
The stopping distance is proportional to the square of the speed of the car. This is because the braking force needed to bring the vehicle to a stop is constant. Thus, if the speed doubles, the stopping distance increases by a factor of four.
2Step 2: Use the Formula for Stopping Distance
We have the stopping distance formula:\[s = \frac{v^2}{2a}\]where \(s\) is the stopping distance, \(v\) is the speed, and \(a\) is the deceleration. We know that the stopping distance at \( 60 \text{ km/h} \) is 20 meters.
3Step 3: Calculate New Speed Factor
The car is going twice as fast, so the new speed is \(120 \text{ km/h}\), which is factor of 2 of \(60 \text{ km/h}\). Hence, the stopping distance increases by a factor of 2 squared:\[\text{Factor} = 2^2 = 4\]
4Step 4: Find the New Stopping Distance
Multiply the initial stopping distance by the factor of 4 to find the new stopping distance:\[s_{new} = 20 \text{ m} \times 4 = 80 \text{ m}\]
5Step 5: Select the Correct Answer
Based on the calculations, the stopping distance when the car travels at \(120 \text{ km/h}\) is \(80 \text{ m}\). Therefore, the correct choice is (d) \(80 \text{ m}\).
Key Concepts
Stopping DistanceKinematicsSpeed and VelocityBraking Force
Stopping Distance
Stopping distance is the total distance a vehicle travels before it comes to a complete stop after the brakes are applied. It comprises two main components: the reaction distance and the braking distance. Reaction distance is the length your vehicle moves in the time it takes for you to react to a situation and press the brakes. Braking distance is the distance the car travels after the brakes are applied until it stops completely.
The stopping distance is an important safety consideration as it helps determine how much space is needed to bring a vehicle to rest safely. This distance can be influenced by several factors, such as the driver's reaction time, the condition of the vehicle's brakes, weather conditions, and the speed of the vehicle itself.
The stopping distance is an important safety consideration as it helps determine how much space is needed to bring a vehicle to rest safely. This distance can be influenced by several factors, such as the driver's reaction time, the condition of the vehicle's brakes, weather conditions, and the speed of the vehicle itself.
Kinematics
Kinematics is a branch of physics focused on the study of motion without considering the forces that cause motion. When solving physics problems related to stopping distance, kinematics provides tools like the equations of motion, which describe how an object's velocity and position change over time.
In stopping distance problems, like the one presented, we often use these principles to calculate how far a car will travel at a certain speed before stopping. The formulas in kinematics, such as \[s = \frac{v^2}{2a}\]allow calculating stopping distances by relating speed (velocity squared) and deceleration. Understanding kinematics is key to solving problems involving speed, velocity, and stopping distances accurately.
In stopping distance problems, like the one presented, we often use these principles to calculate how far a car will travel at a certain speed before stopping. The formulas in kinematics, such as \[s = \frac{v^2}{2a}\]allow calculating stopping distances by relating speed (velocity squared) and deceleration. Understanding kinematics is key to solving problems involving speed, velocity, and stopping distances accurately.
Speed and Velocity
Speed and velocity are fundamental concepts in physics that describe a moving object. Speed refers to how fast an object is moving and is a scalar quantity with magnitude but no direction. In contrast, velocity is a vector quantity; it represents both the speed and direction of the object's motion.
In the context of the given problem, speed is critically important as it directly influences the stopping distance. The example shows that when the speed is doubled from 60 km/h to 120 km/h, it increases the stopping distance by a factor of four. This relationship arises because speed impacts the amount of kinetic energy a car has, which needs to be dissipated through the brakes to bring the car to a halt. Thus, understanding the distinction and relationship between speed and velocity is crucial in solving kinematics problems.
In the context of the given problem, speed is critically important as it directly influences the stopping distance. The example shows that when the speed is doubled from 60 km/h to 120 km/h, it increases the stopping distance by a factor of four. This relationship arises because speed impacts the amount of kinetic energy a car has, which needs to be dissipated through the brakes to bring the car to a halt. Thus, understanding the distinction and relationship between speed and velocity is crucial in solving kinematics problems.
Braking Force
Braking force is the force applied by the braking system of a vehicle to slow down or stop its motion. It plays a crucial role in determining stopping distance. The force of braking must be sufficient to overcome the kinetic energy of the moving vehicle, bringing it to a rest.
In many problems, the braking force is considered constant. This assumption helps simplify calculations, allowing us to predict how changes in speed affect stopping distances. For instance, doubling the speed requires four times the stopping distance because the kinetic energy, which is proportional to the square of the velocity, must be dissipated by the same constant braking force.
In many problems, the braking force is considered constant. This assumption helps simplify calculations, allowing us to predict how changes in speed affect stopping distances. For instance, doubling the speed requires four times the stopping distance because the kinetic energy, which is proportional to the square of the velocity, must be dissipated by the same constant braking force.
- Higher speed requires more energy to be dissipated.
- The braking force needs to counter this increased energy.
- The constant nature of braking force plays a key role in these calculations.
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