Problem 63 - Master Resource Book in JEE Main Physics

Question

A car is moving with at a constant speed of \(60 \mathrm{kmh}^{-1}\) on a straight road. Looking at the rear view mirror, the driver finds that the car following him is at distance of \(100 \mathrm{~m}\) and is approaching with a speed of \(5 \mathrm{~km} \mathrm{~h}^{-1}\). In order to keep track of the car in the rear, the driver begins to glance alternatively at the rear and side mirror of his car after every \(2 \mathrm{~s}\) till the other car overtakes. If the two cars were maintaining their speeds, which of the following statement (s) is/are correct? (a) The speed of the car in the rear is \(65 \mathrm{~km} \mathrm{~h}^{-1}\) (b) In the side mirror the car in the rear would appear to approach with a speed of \(5 \mathrm{~km} \mathrm{~h}^{-1}\) to the driver of the leading car (c) In the rear view minor, the speed of the approaching car would appeat to decrease as the distance between the. cars decreases (d) In the side mirror, the speed of the approaching car would appear to increase as the distance between the cars decreases

Step-by-Step Solution

Answer

Statements (a) and (b) are correct; statements (c) and (d) are incorrect.

Step 1: Determine the Speed of the Rear Car

The problem states that the car in the rear is approaching the leading car with a relative speed of \(5 \mathrm{~km} \mathrm{~h}^{-1}\). Since the leading car is moving at \(60 \mathrm{~km} \mathrm{~h}^{-1}\), the absolute speed of the rear car is calculated by adding the relative speed to the speed of the leading car:\[\text{Speed of rear car} = 60 + 5 = 65 \mathrm{~km} \mathrm{~h}^{-1}.\] Therefore, the statement (a) is correct.

Step 2: Analyze the Side Mirror Perception

In the side mirror, the motion of the car in the rear with respect to the leading car is perceived. The car in the rear is closing the gap at \(5 \mathrm{~km} \mathrm{~h}^{-1}\), directly matching the relative speed given. Therefore, statement (b) is correct.

Step 3: Examine the Rear View Mirror Perception

In the rear-view mirror, as the distance between the cars decreases, no change occurs in the relative speed. The car in the rear continues to approach at a constant relative speed of \(5 \mathrm{~km} \mathrm{~h}^{-1}\), contradicting statement (c), which implies a perceived decrease in speed. Thus, statement (c) is incorrect.

Step 4: Evaluate the Side Mirror Speed Perception

In the side mirror, regardless of distance, the relative speed remains constant. As the rear car continuously approaches at a speed of \(5 \mathrm{~km} \mathrm{~h}^{-1}\), there is no increase in this perceived speed contrary to the apparent increase implied by statement (d). Hence, statement (d) is incorrect.

Key Concepts

Understanding Constant Speed

When we talk about constant speed, it means that an object is moving at a steady rate without any increase or decrease over time. For example, in this problem, the leading car is traveling at a constant speed of \(60 \, \text{km/h}\). This implies that every hour, the car covers 60 kilometers without any deviation.

Maintaining constant speed is crucial for calculating distances and predicting how long it will take for one object to catch up with another. It’s like setting cruise control on a long journey.

• The constant speed of the leading car ensures predictability in calculations.
• Knowing the speed of the moving objects helps in understanding and calculating the relative speed.

When you know a vehicle's constant speed, it becomes simpler to analyze the scenario or solve problems related to motion.

The Role of the Rear View Mirror

The rear view mirror in a car allows the driver to see what's happening behind them. In this exercise, the driver uses the rear view mirror to monitor the car approaching from behind. This activity is essential for assessing the traffic situation and for safety reasons.

When the car in the rear is seen through the rear view mirror, it doesn't change its relative speed due to perspective. The car continues to approach at the same relative speed, which is \(5 \, \text{km/h}\) in this scenario.

• The rear view mirror offers real-time, albeit limited, information about objects directly behind.
• Using such a mirror can help drivers make informed decisions while driving.
• The distance of the approaching car decreases over time, but its speed relative to the leading car remains unchanged in this context.

Understanding Relative Speed Perception

Relative speed perception is how the speed of a moving object appears to someone else who is also moving, in this case, the driver of the leading car observing the following car.

In this exercise, the car is approaching the leading vehicle at a relative speed of \(5 \, \text{km/h}\). This means that to the driver of the leading car, the car following appears to be closing in at this speed, regardless of their combined actual speeds. This results from the speed difference:

• Calculating relative speed involves subtracting one object's speed from another.
• The perceived speed affects a driver's interpretation and decisions.
• Relative speed simplifies understanding complex motion between two independent moving entities.

If you imagine two objects moving at an identical speed, they appear stationary relative to each other, no matter how fast they are going. However, any difference in their speeds results in relative movement, as observed by someone involved in the motion.