Motion in A Plane

The x and y coordinates of a particle moving in a plane are given be x(t) = acos(pt) and y(t) = bsin(pt) where a, b (<a) and p are positive constants of appropriate dimensions and t is time. Then , which of the following is not true?

• The path of the particle is an ellipse

• Velocity and acceleration of the particle are perpendicular to each other at t = π/2p

• Acceleration of the particle is always directed towards a fixed point

• Distance travelled by the particle in the time interval between t = 0 and t = π/2p is a

A stick of length L and mass M lies on a frictionless horizontal surface on which it is free to move in any way. A ball of mass m moving with speed V collides elastically with the stick as shown in fig below. If after the collision, the ball comes to rest, then what should be the mass of the ball?

• m = 2M

• m = M

• m= M/2

• m = M/A

A man is at height of 100 m. he sees a car which makes an angle of $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$ with man's position. If the car moves to a point where angle is $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ what is the distance moved by it? 

• $\left(\frac{100}{\sqrt{3}}\right)\mathrm{m}$

• $\left(200\sqrt{3}\right)\mathrm{m}$

• $\left(\frac{200}{\sqrt{3}}\right)\mathrm{m}$

• $\left(\frac{150}{\sqrt{3}}\right)\mathrm{m}$

The angular velocity of second hand clock, of a clock is

• $\left(\frac{\mathrm{\pi }}{6}\right)\mathrm{rad} {\mathrm{s}}^{-1}$

• $\left(\frac{\mathrm{\pi }}{60}\right)\mathrm{rad} {\mathrm{s}}^{-1}$

• $\left(\frac{\mathrm{\pi }}{30}\right)\mathrm{rad} {\mathrm{s}}^{-1}$

• $\left(\frac{\mathrm{\pi }}{15}\right)\mathrm{rad} {\mathrm{s}}^{-1}$

The maximum range of a gun on horizontal terrain is 16 km if g=10m/s^2. What must be the muzzle velocity of the shell?

• 200 m/s

• 100 m/s

• 400 m/s

• 300 m/s

If a vector$2 \hat{\mathrm{i}} + 3 \hat{\mathrm{j}} + 8 \hat{\mathrm{k}}$ is perpendicular to the vector $4 \hat{\mathrm{j}} - 4 \hat{\mathrm{i}} + \mathrm{\alpha } \hat{\mathrm{k}}$, then the value of α is

• -1

• $\frac{1}{2}$

• $-\frac{1}{2}$

• 1

A particle starting from the origin (0, 0) moves in a straight line in the (x, y) plane. Its coordinates at a later time are $\left(\sqrt{3}, 3\right)$. The path of the particle makes with the x-axis an angle of

• 30^o

• 45^o

• 60^o

• 0^o

A wheel has angular acceleration of 3.0 rad/s² and an initial angular speed of 2.00 rad/s. In a time of 2 s it has rotated through an angle (in radian) of

• 6

• 10

• 12

• 4

Under the influence of a uniform magnetic field a charged particle is moving in a circle of radius  R with constant speed v. The time period of the motion

• depends on v and not on R

• depends on both R and v

• is independent of both R and v

• depends on R and not on v

A projectile is thrown in the upward direction making an angle of 60° with the horizontal direction with a velocity of 147 ms^-1 Then the time after which its inclination with the horizontal is 45°, is

• 15 s

• 10.98 s

• 5.49 s

• 2.745 s