# Kinematics Problems

1. The displacement of an object is given by the following equation.
$x=2t+t^{2}_{}-3t^{3}$

Calculate the velocity of the object when its acceleration is zero.

We have, $x=2t+t^{2}_{}-3t^{3}$

Or, velocity $v=\frac{dx}{dt}=\frac{d}{dt}(2t+t^{2}_{}-3t^{3})$

Or, $v=2+2t-9t^{2}$\ldots \ldots \ldots \ldots \ldots (1)

Acceleration, $a=$$\frac{dv}{dt}=\frac{d}{dt}(2+2t-9t^{2}) Or, a=2-18t Or, 2-18t=0, as acceleration is zero Or, t=2/18=1/9 Putting this value in equation (1), we get, v=2+2\times \frac{1}{9}-9(1/9)^{2} Or, v=2+\frac{2}{9}-\frac{1}{9} Or, v=19/9. 2. An object is released from a certain height under the influence of gravity. State the nature of the velocity-displacement graph. For an object undergoing free fall, the equation of motion is given by, x=ut+1/2 gt^{2}=1/2 gt^{2}, initial velocity u = 0 velocity v=\frac{dx}{dt}=\frac{d}{dt}(1/2 gt^{2}) Or, v=gt Or, t=v/g Putting the value of t in the above equation, we get, x=1/2 g \frac{v^{2}}{g^{2}} Or, v^{2}=2gx This is an equation of parabola. Therefore, the velocity-displacement graph gives a parabolic path. 3. An object is released from a height H. During the journey, it covers H/2distance in time t_{1} and the remaining distance in time t_{2}. Prove that t_{1}$$>$ $t_{2}$.

For the first half, we have the equations of motion,

$H/2=1/2 gt_{1}^{2}$ ……………. (1)

$v=gt_{1}^{}$ ……………… (2)

For the second half,

$H/2=vt_{2}+1/2 gt_{2}^{2}$

Or, $H/2=gt_{1}t_{2}+1/2 gt_{2}^{2}$ …………….. (3)

Comparing equations (1) and (3), we get,

$1/2 gt_{1}^{2}=gt_{1}t_{2}+1/2 gt_{2}^{2}$

Or, $t_{1}^{2}=2 t_{1}t_{2}+t_{2}^{2}$

Or, $2 t_{1}t_{2}+t_{2}^{2}-t_{1}^{2}=0$

Applying the formula of quadratic equation, we get,

Therefore, $t_{1}>t_{2}$.

4. A stone of mass 0.01kg is released from the terrace of a building of height 70 m. Simultaneously another stone of mass 0.008kg is thrown up from the same place with a speed of 30 m/s. Calculate their relative speed.

For stone of mass 0.01kg, we apply kinematic equation,

$v_{1}=0-gt=-gt$

For a stone of mass 0.008kg, we apply the kinematic equation

$v_{2}=30-gt$

Relative velocity of first stone with respect to second stone,

$v_{12}=-gt-(30-gt)=-30$m/s

Similarly, $v_{21}=30-gt-(-gt)=30$m/s

So, the relative velocity is 30 m/s.

5. A racing car moves along a circular path of radius R at speed v so that it makes one revolution in every T seconds. If the car moves around a different circular path of radius 4R so that it takes one revolution in every T/2 seconds, then find the ratio of the magnitude of centripetal acceleration.

We have centripetal acceleration, $a=V^{2}/R$

Now, $V=2\pi R/T$

So, for the first track, centripetal acceleration $a_{1}^{}=\frac{4\pi ^{2}R^{2}}{RT^{2}}=\frac{4\pi ^{2}R^{}}{T^{2}}$

For the second track, $V=2\pi 4R/(T/2)=16\pi R/T$

For the second track, centripetal acceleration $a_{2}^{}=\frac{16\times 16\pi ^{2}R^{2}}{4RT^{2}}$

Or, $a_{2}^{}=\frac{64\pi ^{2}R^{}}{T^{2}}=16 a_{1}$

Or, $a_{1}$/$a_{2}^{}$=1/16

Or, $a_{1}:a_{2}=1:16$.

6. A bus starts from rest and accelerates uniformly over a time of 6 seconds for a distance of 110 m. Calculate the acceleration of the car.

We have, initial velocity, $u=0$, $t=6s$, $s=110m$, $a=?$

Using kinematic equation,

$s=ut+1/2 at^{2}$

Or, $110=0+1/2 (a\times 36)$

Or, $36a=220$

Or, $a=6.11 m/s^{2}$.

7. Four cyclists A, B, C and D are in motion. Their relative velocities are given below:

= 20 m/s towards north

= 20 m/s towards east

= 20 m/s towards south

Calculate .

We have,

……….(1)

…………..(2)

………….(3)

Subtracting equation (2) from equation (1), we get,

……………..(4)

Adding equation (3) and (4), we get

Or, = $-20{i}$

So, = 20 m/s towards west.

8. The displacement (in meter) of an object moving along x-axis is given by $x=9t+4 t^{2}$. Calculate Instantaneous velocity at $t=2s$.

$v=dx/dt=\frac{d}{dt}($$9t+4 t^{2}$)

Or, $v=9+8t$

Instantaneous velocity at $t=2s$is $v=9+8\times 2=25$m/s.