Relative Motion

Step 1: Vertical component of velocity at height 5 m is Step 2: Use vertical motion equation: Conclusion: is the maximum height.

Let the student run with constant speed , and the bus starts from rest with acceleration . Let the time taken to meet be . - Distance travelled by student: - Distance travelled by bus (starting from rest): Since initial separation is 16 m: Multiply both sides by 2: For student to just catch the bus, the equation should have only one solution ⇒ Discriminant

Components of : Total y-force = , Resultant force: Acceleration

Let rain’s velocity with respect to the car be .
Let horizontal component = car's speed = 40 km/h, and angle with vertical is .
Using
Now, actual rain velocity = km/h.
So with respect to car: magnitude of relative velocity =

Let the time taken for the car to cross the truck be . - Distance traveled by the truck in time is: - Distance traveled by the car in time with an initial velocity of 0 and acceleration is: Since the car needs to cross the truck, the distance traveled by the car should be equal to the distance traveled by the truck plus the length of the truck. We assume the truck's length to be negligible. Thus, the equation becomes: Solving for , we get: Now, substituting into the equation for the distance traveled by the car: Thus, the distance the car has to travel to cross the truck again is . The correct answer is option (3), .

To minimize crossing time, swimmer must maximize velocity component perpendicular to river: .
This happens when swimming directly across, i.e., perpendicular to river flow.
Time to cross: (minimum).
Downstream drift will occur, but crossing time is minimized.

Using the principle of conservation of momentum, we can calculate the total distance the girl moves relative to the ground. The velocity of the plank and the girl are related by the equation: Thus, the total displacement of the girl relative to the ground is:
So, the girl moves 3.2 meters relative to the ground.

Step 1: Define variables.
Distance =
Boat speed in still water kmh
Stream speed kmh
Total time hours

Step 2: Calculate upstream and downstream speeds.
Upstream speed kmh
Downstream speed kmh
Step 3: Express time for upstream and downstream travel.
Time upstream
Time downstream
Step 4: Use total time.


Step 5: Solve for .



km
Step 6: Conclusion.
The distance travelled upstream (before returning) is 240 km.