Analyzing the Motion of Cars P and Q
This problem involves calculating the time car Q stopped, given information about the speeds, directions, and distances covered by two cars, P and Q, starting from the same point X.
Calculating Car P's Distance
Car P starts at 10 AM and travels North at a constant speed ($v_P$) of 25 km/h. The time considered is 11:30 AM. The total time duration ($t_{total}$) is from 10:00 AM to 11:30 AM, which is 1 hour and 30 minutes, or 1.5 hours. Car P travels continuously for this entire duration. The distance covered by Car P ($d_P$) is calculated using the formula: $d = \text{speed} \times \text{time}$ So, $d_P = v_P \times t_{total}$ $d_P = 25 \text{ km/h} \times 1.5 \text{ h}$ $d_P = 37.5 \text{ km}$ Since Car P travels North from point X, its distance from X at 11:30 AM ($D_P$) is 37.5 km.
Calculating Car Q's Travel and Stop Time
Car Q also starts from point X at 10 AM and travels East with a speed ($v_Q$) of 30 km/h. Car Q stops for some duration ($t_{stop}$) during its journey. The total time elapsed until 11:30 AM is still $t_{total} = 1.5$ hours. Let the actual time Car Q spent travelling be $t_{travel\_Q}$. The total time is the sum of the travel time and the stop time: $t_{total} = t_{travel\_Q} + t_{stop}$ $1.5 \text{ h} = t_{travel\_Q} + t_{stop}$ The problem states that Car Q stops *after* travelling for one hour. This implies that the total time Car Q travelled, $t_{travel\_Q}$, must be greater than or equal to 1 hour. The distance covered by Car Q ($d_Q$) is: $d_Q = v_Q \times t_{travel\_Q}$ $d_Q = 30 \text{ km/h} \times t_{travel\_Q}$ Since Car Q travels East from point X, its distance from X at 11:30 AM ($D_Q$) is $d_Q$.
Equating Distances and Solving for Travel Time
We are given that at 11:30 AM, both cars are at the same distance from X. Therefore, $D_P = D_Q$. $37.5 \text{ km} = 30 \text{ km/h} \times t_{travel\_Q}$ To find the travel time of Car Q, we rearrange the equation: $t_{travel\_Q} = \frac{37.5 \text{ km}}{30 \text{ km/h}}$ $t_{travel\_Q} = 1.25 \text{ h}$ This travel time (1.25 hours) is indeed greater than 1 hour, satisfying the condition that Car Q stopped after travelling for at least one hour.
Determining Car Q's Stop Duration
Now we can find the time Car Q stopped using the total time equation: $t_{stop} = t_{total} - t_{travel\_Q}$ $t_{stop} = 1.5 \text{ h} - 1.25 \text{ h}$ $t_{stop} = 0.25 \text{ h}$
Converting Stop Time to Minutes
The question asks for the duration in minutes. To convert hours to minutes, we multiply by 60: $t_{stop} (\text{in minutes}) = 0.25 \text{ h} \times 60 \text{ minutes/h}$ $t_{stop} (\text{in minutes}) = 15 \text{ minutes}$
Final Answer
Car Q stopped for 15 minutes.
Important Questions from Speed Distance and Time
In a 500 m race, P and Q have speeds in the ratio of 3: 4. Q starts the race when P has already covered 140 m.
What is the distance between P and Q (in m) when P wins the race?A vehicle is moving at a speed of 12 m/s on a level road. It applies emergency brakes and starts to skid without rolling in a straight path. The deceleration of the vehicle is constant after braking and it comes to rest at a distance of 15 m. Assuming, $g = 10$ m/s$^2$, the coefficient of kinetic friction between the tyres and road is _________ [round off to 2 decimal places]
- The distance between Delhi and Agra is 233 km. A car P started travelling from Delhi to Agra and another car Q started from Agra to Delhi along the same road 1 hour after the car P started. The two cars crossed each other 75 minutes after the car Q started. Both cars were travelling at constant speed. The speed of car P was 10 km/hr more than the speed of car Q. How many kilometers the car Q had travelled when the cars crossed each other?
Two trains started at 7AM from the same point. The first train travelled north at a speed of 80km/h and the second train travelled south at a speed of 100 km/h. The time at which they were 540 km apart is _______________ AM.
- From the time the front of a train enters a platform, it takes 25 seconds for the back of the train to leave the platform, while travelling at a constant speed of 54 km/h. At the same speed, it takes 14 seconds to pass a man running at 9 km/h in the same direction as the train. What is the length of the train and that of the platform in meters, respectively?
