Frame of reference, motion in a straight line, speed and velocity
- In some appropriate units, time (t) and position (x) relation of a moving particle is given by t = x^2 + x. The acceleration of the particle is Answer: 2. -2/(2x+1)³ Key idea: differentiate t(x) to get velocity, then use a=v dv/dx . dt/dx=2x+1⇒ v=dx/dt=1/2x+1 a=vdv/dx=1/2x+1·(-2/(2x+... 1
- A vehicle travels half the distance with speed v and the remaining distance with speed 2v. Its average speed is: Answer: 4. 4v/3 Average speed = total distance / total time. Let the total distance be 2s . t₁=s/v , t₂=s/2v , total time =3s/2v . v_avg=2s/... 1
- The x and y coordinates of the particle at any time are x = 5t - 2t^2 and y = 10t respectively, where x and y are in meters and t in seconds. The acceleration of the particle at t = 2s is : Answer: 2. - 4 m/s ² Key idea: acceleration components are second derivatives of position. a_x=d²x/dt²=-4 m/s ² , a_y=d²y/dt²=0 . Accelerati... 1
- Two cars P and Q start from a point at the same time in a straight line and their positions are represented by x_P(t)=at+bt^2 and x_Q(t)=ft-t^2. At what time do the cars have the same velocity? Answer: 3. f-a/2(1+b) Velocity is the derivative of position; equate the two velocities. v_P=a+2bt ; v_Q=f-2t . a+2bt=f-2t⇒ 2t(b+1)=f-a . t=... 1
- If the velocity of a particle is v = At + Bt^2, where A and B are constants, then the distance travelled by it between 1s and 2s is : Answer: 3. 3/2A + 7/3B Key idea: distance is the integral of speed over time (here v≥0 ). s=∫₁²(At+Bt²)dt=A4-1/2+B8-1/3=3/2A+7/3B . 1