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i want some imp questions on the chapter motion in a straight line

1. If the displacement of a body is proportional to square of time, state whether the body is moving with uniform velocity or uniform acceleration. 

Suppose the displacement of the body is given by x = kt² where K is constant of proportionality.

                     
As the acceleration is independent of time, the body is moving with uniform acceleration.

2. A stone is dropped into a well of depth 196 m.  The sound of the splash is heard after 6.88s.  Find the velocity of sound in air. ( g = 9.8 m/s²)

Here, S = 196 m,  a = 9.8 m/s², V₀ = 0

Time taken by the stone to reach the bottom can be calculated as follows.

The sound is heard after 6.88 s.

Therefore Time taken by sound to reach the top of the well
            = 6.88 - 6.325 = 0.555 s.

3. A car travels a distance A to B at a speed  of 40 km/h and returns to A at a speed of 30 km/h.

(i) What is the average speed for the whole Journey? 

Let AB = S, time taken to go from A to B,  t₁ = S/40 h

              and time taken to go from B to A, t₂ = S/30 h

              Total time taken = t₁ + t₂

(ii) What is the average velocity? 

 Total displacement = zero, since the car returns to the original position.

4. A car is running at a steady speed of 72 km/h.  Its speed is reduced to 18 km/h in just 2 sec. on applying brakes.

(i) Find  Its retardation.

  
(ii) V₀ = 20 m/s, v₁ = 0, a = - 75 m/s^2,  t = ?

v₁ = v₀ + at 

5. A person travelled a distance of 200 km between two cities by a car covering the first half distance of the journey at a constant speed of 40 km/h and the remaining half distance at a constant speed of v km/h. If the average speed of the person for the entire journey is 48 kmh/h, find v.

6. A train 100 m long is moving with a uniform velocity of 45 km/h. What is the time it will take to cross a bridge 1 km long?
Total distance to be covered
                 = Length of bridge + Length of train = 1 + 0.1 = 1.1 km
Therefore Time taken by the train to cross the bridge is

7. Two cars started simultaneously towards each other from towns A and B which are 480 km apart. It took first car travelling from A to B 8 hours to cover the distance meet after starting and at what distance from town A. Assume that both the cars travel at constant speed.

Speed of car travelling from town A is
             V_A = ⁴⁸⁰/₈ = 60 km/h
Speed of car travelling from town B is
            V_B = ⁴⁸⁰/₁₂ = 40 km/h
Let the two cars meet after time t hours. In this time, the distance travelled from a is 60t and the distance travelled from B is 40 t. When the cars meet, the total distance travelled = 480 km.

The distance from town A at the meeting point

                        = V_A × t = 60 × 4.8 = 288 km

8. A body is moving in a straight line along X-axis. Its distance x in metres from the origin is given by ; x = 8 t – 3 t²
The time t is in seconds. Find the average speed of the body in time interval t = 0 to t = 1 s.

9. A ball thrown up is caught by the thrower after 4 seconds. How high did it go and with what velocity was it thrown? How far below its highest point was it 3 seconds after start?
  The time for upward journey is equal to the time for the downward journey. Therefore, time taken by the ball to reach the highest point = 2 sec. At the highest point v = 0.

Therefore            v = v₀ + g t = v₀ - 9.8 × 2

or                        0 = v₀ - 9.8 × 2     therefore  v₀ = 10.6 ms^-1

The ball takes 2 seconds to reach the highest point.   Therefore, to find height below the highest point, t = 3 – 2 = 1s.

10. A rifle bullet loses ¹/₂₀ of its velocity in passing through a plank. What is the least number of planks required to stop the bullet?
Let                             S = thickness of the plank
                                 = uniform retardation of the plank
                                v₀ = initial velocity of the bullet
The velocity of he bullet after passing through a plank =  

   
   Therefore Maximum  number of planks required = 10 + 1 = 11.