The position of a particle is given by
`r = 3.0t hati − 2.0t 2 hatj + 4.0 hatk m`
Where t is in seconds and the coefficients have the proper units for r to be in metres.
(a) Find the v and a of the particle?
(b) What is the magnitude and direction of velocity of the particle at t = 2.0 s?
(a)vecv(t) = (3.0 hati - 4.0t hatj); veca = -4.0 hatj
The position of the particle is given by:
`vecr =3.0t hati - 2.0t^2 hatj + 4.0 hatk`
Velocity `vecv`, of the particle is given as:
`vecv = (dvecr)/(dt) = d/(dt)(3.0t hati - 2.0t^2 hatj + 4.0 hatk)`
`:.vecv = 3.0 hati - 4.0t hatj`
Acceleration `veca`, of the particle is given as:
`veca = (dvecv)/(dt) = d/(dt)(3.0 hati - 4.0t hatj)`
`:.veca = -4.0 hatj`
(b) 8.54 m/s, 69.45° below the x-axis
We have velocity `vecv = 3.0 hati - 4.0t hatj`
At t = 2.0 s:
`vecv = 3.0 hati - 8.0 hatj`
The magnitude of velocity is given by:
|vecv|=sqrt(362+(-8)^2) = sqrt73 = 8.54 m/s
Direction, `theta = tan^(-1)(v_y)/v_x)`
= `tan^(-1)(-8/3) = -tan^(-1)(2.667)`
The negative sign indicates that the direction of velocity is below the x-axis.
Here `vecr(t) = (3.0t hati - 2.0t^2 hatj + 4.0 hatk)m`
(a) `vecv(t) = vec(dr)/(dt) = (3.0 hati - 4.0t hatj) "m/s"`
and `veca(t) = vec(dt)/(dt) = (-4.0 hatj)"m/s"^2`
(b) Magnitude of velocity at t= 2.0 s
`v_(t = 2s) = sqrt((3.0)^2+(-4.0xx2)^2) = sqrt(9+64) = sqrt(73)`
= 8.54 `ms^(-1)`
THis velocity will subtend an angle `beta` from x-axis, where `tan beta =((-4.0xx2))/(3.0)` = -2.667
`:.beta = tan^(-1)(-2.6667) =- -69.44^@ = 69.44^@` from negative x-axis`
A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. If the round trip takes 10 min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist?
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?
Read each statement below carefully and state, with reasons, if it is true or false:
The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector
For any arbitrary motion in space, which of the following relations are true:
a) `V_"average"` = (1/2) (v (t1) + v (t2))
b) `V_"average"` = [r(t2) - r(t1) ] /(t2 – t1)
c) v (t) = v (0) + a t
d) r (t) = r (0) + v (0) t + (1/2) a t2
e) `a_"average"` = =[ v (t2) - v (t1 )] /( t2 – t1)
(The ‘average’ stands for average of the quantity over the time interval t1 to t2)