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
- A car is moving on a horizontal surface in a straight line with a constant velocity of 3 m/s. A ball is thrown vertically upwards at time $t = 0$ from the top of the moving car with a velocity of 20 m/s. The acceleration due to gravity is 10 m/s$^2$.
At what value(s) of time $t$ in second(s), the ball is at a height of 15 m from the top of the moving car? - Velocity of an object fired directly in upward direction is given by $V = 80 – 32 \ t$, where $t$ (time) is in seconds. When will the velocity be between 32 m/sec and 64 m/sec?
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?- An automobile travels from city A to city B and returns to city A by the same route. The speed of the vehicle during the onward and return journeys were constant at 60 km/h and 90 km/h, respectively. What is the average speed in km/h for the entire journey?
Two cars start at the same time from the same location and go in the same direction. The speed of the first car is 50 km/h and the speed of the second car is 60 km/h. The number of hours it takes for the distance between the two cars to be 20 km is ___________.
