Advertisements
Advertisements
प्रश्न
On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
Advertisements
उत्तर
(i) The path followed by the motorist will be a closed hexagonal path.
Suppose the motorist starts his journey from the , point O. He takes the turn at the point C.
Displacement = `vec(OC)`
Here OC = `sqrt((OB)^2+(BC)^2) =sqrt((OF+FB)^2+(BC)^2)`
`=sqrt((500 cos30^@ + 500 cos 30^@)^2 + (500)^2)`
`=sqrt((2xx500xxsqrt3/2)^2+(500)^2)`
`=500sqrt4 = 1000 m = 1 km`
Total path length = 500 m + 500 m = 1500 m = 1.5 km
`"Magnitude of displacement"/"Total path length" = 1/1.5 = 2/3 = 0.67`
(ii) After completing six turns, the motorist returns to the starting point, creating a closed hexagonal path.
The displacement is zero since the start and end points coincide |`vec("OC")`| = 0`
The total path length is 6 × 500 = 3,000 metres
Ratio of magnitude of displacement and path length is zero.
(iii) At the eighth turn, the motorist reaches point B. The displacement is the straight-line distance from the starting point O to point B.
Since ∠BOA = ∠OBA = 30°, by trigonometry, the horizontal and vertical components of the displacement are equal to OA cos(30°) and AB cos(30°), respectively.
The total displacement |`vec("OB")`| is twice the horizontal component ON3, because OA = AB, each being 500 m. Thus:
|`vec("OB")`| = 2 × 500 × cos(30°)
= `1000xxsqrt(3)/2 = 500sqrt(3) "m"`
= 0.866 km
The total path length after eight turns is 8 × 500 = 4000 m
`("|"vec("OB")"|")/"Path Length" = (500sqrt(3))/4000`
= `sqrt(3)/8`
APPEARS IN
संबंधित प्रश्न
In a harbour, wind is blowing at the speed of 72 km/h and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of 51 km/h to the north, what is the direction of the flag on the mast of the boat?
For any arbitrary motion in space, state whether the following statement is true:
`"V"_"average"` = `(1/2)("v"("t"_1) + "v"("t"_2))`
(The ‘average’ stands for average of the quantity over the time interval t1 to t2)
For any arbitrary motion in space, state whether the following statement is true:
`"V"_"average"` = [r(t2) - r(t1) ] /(t2 – t1)
For any arbitrary motion in space, state whether the following statement is true:
v (t) = v (0) + a t
(The ‘average’ stands for average of the quantity over the time interval t1 to t2)
For any arbitrary motion in space, state whether the following statement is true:
`"r"("t") = "r"(0) + "v"(0)"t" + 1/2 "a" "t"^2 `
(The ‘average’ stands for average of the quantity over the time interval t1 to t2)
In a two dimensional motion, instantaneous speed v0 is a positive constant. Then, which of the following are necessarily true?
In a two dimensional motion, instantaneous speed v0 is a positive constant. Then which of the following are necessarily true?
Two balls A and B are placed at the top of 180 m tall tower. Ball A is released from the top at t = 0 s. Ball B is thrown vertically down with an initial velocity 'u' at t = 2 s. After a certain time, both balls meet 100 m above the ground. Find the value of 'u' in ms-1. [use g = 10 ms -2]:
A small toy starts moving from the position of rest under constant acceleration. If it travels a distance of 10 minutes, the distance travelled by the toy in the next will be ______.
Two vectors `vec"A"` and `vec"B"` have equal magnitudes. If magnitude of `vec"A"` + `vec"B"` is equal to two times the magnitude of `vec"A"` - `vec"B"`, then the angle between `vec"A"` and `vec"B"` will be :
A person can throw a ball up to a maximum range of 100 m. How high above the ground he can throw the same ball?
The position of a particle is x - y plane is described by the variables x = at3 and y = 2at. Then the acceleration of the particle ______.
A stone is dropped from the top of the tower and travels 24.5 m in the last second of its journey. The height of the tower is ______.
(g = 9.8 m/s2)
The displacement s of a particle depends on time t according to the following relation `s = 1/3t^3 - t^2 + t`. The velocity and displacement of the particle at the instant when its acceleration is zero, are respectively ______.
