Physics · Textbook solutions

Motion in a Plane

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 35 questions

3. Motion in a Plane — worked examples

9 q

Solved Examples

Worked · 9
  1. Solved Ex.3.1
    A person walks from point P to point Q along a straight road in 10 minutes, then turns back and returns to point R which is midway between P and Q after further 4 minutes. If PQ is 1 km, find the average speed and velocity of the person in going from P to R.
  2. Solved Ex.3.2
    A stone is thrown vertically upwards from the ground with a velocity 15 m/s. At the same instant a ball is dropped from a point directly above the stone from a height of 30 m. At what height from the ground will the stone and the ball meet and after how much time? (Use g=10g = 10 m/s2^2 for ease of calculation).
  3. Solved Ex.3.3
    An aeroplane A, is travelling in a straight line with a velocity of 300 km/hr with respect to Earth. Another aeroplane B, is travelling in the opposite direction with a velocity of 350 km/hr with respect to Earth. What is the relative velocity of A with respect to B? What should be the velocity of a third aeroplane C moving parallel to A, relative to the Earth if it has a relative velocity of 100 km/hr with respect to A?
  4. Solved Ex.3.4
    The position vectors of three particles are given by x⃗1=(5i^+5j^)\vec{x}_{1} = (5\hat{i} + 5\hat{j}) m, x⃗2=(5t i^+5t j^)\vec{x}_{2} = (5t\,\hat{i} + 5t\,\hat{j}) m and x⃗3=(5t i^+10t2 j^)\vec{x}_{3} = (5t\,\hat{i} + 10t^{2}\,\hat{j}) m as a function of time tt. Determine the velocity and acceleration for each, in SI units.
  5. Solved Ex.3.5
    The initial velocity of an object is u⃗=5i^+10j^\vec{u} = 5\hat{i} + 10\hat{j} m/s. Its constant acceleration is a⃗=2i^+3j^\vec{a} = 2\hat{i} + 3\hat{j} m/s2^{2}. Determine the velocity and the displacement after 5 s.
  6. Solved Ex.3.6
    An aeroplane is travelling northward with a speed of 300 km/hr with respect to the Earth, when the wind is blowing from east to west at a speed of 100 km/hr. What is the velocity of the aeroplane with respect to the wind?
  7. Solved Ex.3.7
    A stone is thrown with an initial velocity components of 20 m/s along the vertical, and 15 m/s along the horizontal direction. Determine the position and velocity of the stone after 3 s. Determine the maximum height that it will reach and the total distance travelled along the horizontal on reaching the ground. (Assume g=10g = 10 m/s2^2)
  8. Solved Ex.3.8
    An object of mass 50 g moves uniformly along a circular orbit with an angular speed of 5 rad/s. If the linear speed of the particle is 25 m/s, what is the radius of the circle? Calculate the centripetal force acting on the particle.
  9. Solved Ex.3.9
    An object is travelling in a horizontal circle with uniform speed. At t=0t = 0, the velocity is given by u⃗=20i^+35j^\vec{u} = 20\hat{i} + 35\hat{j} km/s. After one minute the velocity becomes v⃗=−20i^−35j^\vec{v} = -20\hat{i} - 35\hat{j}. What is the magnitude of the acceleration?

Exercises

26 q

Choose the correct option

Practice · 5
  1. 1. Choose the correct option.
    Ex Q.1 (i)
    An object thrown from a moving bus is on example of
    1. A.
      Uniform circular motion
    2. B.
      Rectilinear motion
    3. C.
      Projectile motion
    4. D.
      Motion in one dimension
  2. Ex Q.1 (ii)
    For a particle having a uniform circular motion, which of the following is constant
    1. A.
      Speed
    2. B.
      Acceleration
    3. C.
      Velocity
    4. D.
      Displacement
  3. Ex Q.1 (iii)
    The bob of a conical pendulum under goes
    1. A.
      Rectilinear motion in horizontal plane
    2. B.
      Uniform motion in a horizontal circle
    3. C.
      Uniform motion in a vertical circle
    4. D.
      Rectilinear motion in vertical circle
  4. Ex Q.1 (iv)
    For uniform acceleration in rectilinear motion which of the following is not correct?
    1. A.
      Velocity-time graph is linear
    2. B.
      Acceleration is the slope of velocity time graph
    3. C.
      The area under the velocity-time graph equals displacement
    4. D.
      Velocity-time graph is nonlinear
  5. Ex Q.1 (v)
    If three particles A, B and C are having velocities v⃗A\vec{v}_A, v⃗B\vec{v}_B and v⃗C\vec{v}_C which of the following formula gives the relative velocity of A with respect to B
    1. A.
      v⃗A+v⃗B\vec{v}_A + \vec{v}_B
    2. B.
      v⃗A−v⃗C+v⃗B\vec{v}_A - \vec{v}_C + \vec{v}_B
    3. C.
      v⃗A−v⃗B\vec{v}_A - \vec{v}_B
    4. D.
      v⃗C−v⃗A\vec{v}_C - \vec{v}_A

Answer the following questions

Practice · 10
  1. 2. Answer the following questions.
    Ex Q.2 (i)
    Separate the following in groups of scalar and vectors: velocity, speed, displacement, work done, force, power, energy, acceleration, electric charge, angular velocity.
  2. Ex Q.2 (ii)
    Define average velocity and instantaneous velocity. When are they same?
  3. Ex Q.2 (iii)
    Define free fall.
  4. Ex Q.2 (iv)
    If the motion of an object is described by x=f(t)x = f(t) write formulae for instantaneous velocity and acceleration.
  5. Ex Q.2 (v)
    Derive equations of motion for a particle moving in a plane and show that the motion can be resolved in two independent motions in mutually perpendicular directions.
  6. Ex Q.2 (vi)
    Derive equations of motion graphically for a particle having uniform acceleration, moving along a straight line.
  7. Ex Q.2 (vii)
    Derive the formula for the range and maximum height achieved by a projectile thrown from the origin with initial velocity u⃗\vec{u} at an angel θ\theta to the horizontal.
  8. Ex Q.2 (viii)
    Show that the path of a projectile is a parabola.
  9. Ex Q.2 (ix)
    What is a conical pendulum? Show that its time period is given by 2πlcos⁡θg2\pi\sqrt{\frac{l\cos\theta}{g}}, where ll is the length of the string, θ\theta is the angle that the string makes with the vertical and gg is the acceleration due to gravity.
  10. Ex Q.2 (x)
    Define angular velocity. Show that the centripetal force on a particle undergoing uniform circular motion is −mω2r⃗-m\omega^2\vec{r}.

Solve the following problems

Practice · 11
  1. 3. Solve the following problems.
    Ex Q.3 (i)
    An aeroplane has a run of 500 m to take off from the runway. It starts from rest and moves with constant acceleration to cover the runway in 30 sec. What is the velocity of the aeroplane at the take off?
  2. Ex Q.3 (ii)
    A car moving along a straight road with a speed of 120 km/hr, is brought to rest by applying brakes. The car covers a distance of 100 m before it stops. Calculate (i) the average retardation of the car (ii) time taken by the car to come to rest.
  3. Ex Q.3 (iii)
    A car travels at a speed of 50 km/hr for 30 minutes, at 30 km/hr for next 15 minutes and then 70 km/hr for next 45 minutes. What is the average speed of the car?
  4. Ex Q.3 (iv)
    A velocity-time graph is shown in the adjoining figure. [Figure: a velocity-time graph. The vertical axis is labelled "v m/s" and carries the marks 5, 10, 15 and 20; the horizontal axis is labelled "time (s)" and carries the marks 1 to 8. The origin is marked O. The graph consists of three straight segments: a line rising from the origin to the point A at t=3t = 3 s, v=20v = 20 m/s; a horizontal line from A to the point B at t=6t = 6 s, v=20v = 20 m/s; and a line falling from B to the point C, which lies on the time axis at t=8t = 8 s. A dashed horizontal line joins the mark 20 on the velocity axis to A. The foot of the perpendicular dropped from A to the time axis is labelled E (at t=3t = 3 s) and the foot of the perpendicular dropped from B to the time axis is labelled D (at t=6t = 6 s).] Determine: (i) initial speed of the car (ii) maximum speed attained by the car (iii) part of the graph showing zero acceleration (iv) part of the graph showing constant retardation (v) distance travelled by the car in first 6 sec.
  5. Ex Q.3 (v)
    A man throws a ball to maximum horizontal distance of 80 meters. Calculate the maximum height reached.
  6. Ex Q.3 (vi)
    A particle is projected with speed v0v_0 at angle θ\theta to the horizontal on an inclined surface making an angle ϕ\phi (ϕ<θ)(\phi < \theta) to the horizontal. Find the range of the projectile along the inclined surface.
  7. Ex Q.3 (vii)
    A metro train runs from station A to B to C. It takes 4 minutes in travelling from station A to station B. The train halts at station B for 20 s. Then it starts from station B and reaches station C in next 3 minutes. At the start, the train accelerates for 10 sec to reach the constant speed of 72 km/hr. The train moving at the constant speed is brought to rest in 10 sec. at next station. (i) Plot the velocity-time graph for the train travelling from the station A to B to C. (ii) Calculate the distance between the stations A, B and C.
  8. Ex Q.3 (viii)
    A train is moving eastward at 10 m/sec. A waiter is walking eastward at 1.2 m/sec; and a fly is flying toward the north across the waiter's tray at 2 m/s. What is the velocity of the fly relative to Earth
  9. Ex Q.3 (ix)
    A car moves in a circle at the constant speed of 50 m/s and completes one revolution in 40 s. Determine the magnitude of acceleration of the car.
  10. Ex Q.3 (x)
    A particle moves in a circle with constant speed of 15 m/s. The radius of the circle is 2 m. Determine the centripetal acceleration of the particle.
  11. Ex Q.3 (xi)
    A projectile is thrown at an angle of 30∘30^\circ to the horizontal. What should be the range of initial velocity (u) so that its range will be between 40 m and 50 m? Assume g=10 m s−2g = 10\ \text{m s}^{-2}.