<< Chapter < Page Chapter >> Page >

a = | a | = k v y = k v 0

Motion of a balloon

The acceleration of the balloon has two components in mutually perpendicular directions.

Thus, we see that total acceleration is not only one dimensional, but constant as well. However, this does not mean that component accelerations viz tangential and normal accelerations are also constant. We need to investigate their expressions. We can obtain tangential acceleration as time rate of change of the magnitude of velocity i.e. the time rate of change of speed. We, therefore, need to first know an expression of the speed. Now, speed is :

v = k y 2 + v 0 2

Differentiating with respect to time, we have :

a T = đ v đ t = 2 k 2 y 2 k 2 y 2 + v 0 2 x d y d t

a T = d v d t = k 2 y v 0 k 2 y 2 + v 0 2

In order to find the normal acceleration, we use the fact that total acceleration is vector sum of two mutually perpendicular tangential and normal accelerations.

a 2 = a T 2 + a N 2

a N 2 = a 2 a T 2 = k 2 v 0 2 k 4 y 2 v 0 2 k 2 y 2 + v 0 2

a N 2 = k 2 v 0 2 { 1 k 2 y 2 k 2 y 2 + v 0 2 }

a N 2 = k 2 v 0 2 { k 2 y 2 + v 0 2 k 2 y 2 k 2 y 2 + v 0 2 }

a N 2 = k 2 v 0 4 k 2 y 2 + v 0 2

a N = k v 0 2 k 2 y 2 + v 0 2

Nature of motion

Problem : The coordinates of a particle moving in a plane are given by x = A cos(ωt) and y = B sin (ωt) where A, B (<A) and “ω” are positive constants of appropriate dimensions. Prove that the velocity and acceleration of the particle are normal to each other at t = π/2ω.

Solution : By differentiation, the components of velocity and acceleration are as given under :

The components of velocity in “x” and “y” directions are :

đ x đ t = v x = - A ω sin ω t

đ y đ t = v y = B ω cos ω t

The components of acceleration in “x” and “y” directions are :

đ 2 x đ t 2 = a x = - A ω 2 cos ω t

đ 2 y đ t 2 = a y = - B ω 2 sin ω t

At time, t = π 2 ω and θ = ω t = π 2 . Putting this value in the component expressions, we have :

Motion along elliptical path

Velocity and acceleration are perpendicular at the given instant.

v x = - A ω sin ω t = - A ω sin π / 2 = - A ω

v y = B ω cos ω t = B ω cos π / 2 = 0

a x = - A ω 2 cos ω t = - A ω 2 cos π / 2 = 0

a y = - B ω 2 sin ω t = - b ω 2 sin π / 2 = - B ω 2

The net velocity is in negative x-direction, whereas net acceleration is in negative y-direction. Hence at t = π 2 ω , velocity and acceleration of the particle are normal to each other.

Got questions? Get instant answers now!

Problem : Position vector of a particle is :

r = a cos ω t i + a sin ω t j

Show that velocity vector is perpendicular to position vector.

Solution : We shall use a different technique to prove as required. We shall use the fact that scalar (dot) product of two perpendicular vectors is zero. We, therefore, need to find the expression of velocity. We can obtain the same by differentiating the expression of position vector with respect to time as :

v = đ r đ t = a sin ω t i + a cos ω t j

To check whether velocity is perpendicular to the position vector, we take the scalar product of r and v as :

r . v = a cos ω t i + a sin ω t j . - a sin ω t i + a cos ω t j

r . v = - a sin ω t cos ω t + a sin ω t cos ω t = 0

This means that the angle between position vector and velocity are at right angle to each other. Hence, velocity is perpendicular to position vector.

Got questions? Get instant answers now!

Displacement in two dimensions

Problem : The coordinates of a particle moving in a plane are given by x = A cos(ω t) and y = B sin (ω t) where A, B (<A) and ω are positive constants of appropriate dimensions. Find the displacement of the particle in time interval t = 0 to t = π/2 ω.

Solution : In order to find the displacement, we shall first know the positions of the particle at the start of motion and at the given time. Now, the position of the particle is given by coordinates :

x = A cos ω t

and

y = B sin ω t

At t = 0, the position of the particle is given by :

x = A cos ω x 0 = A cos 0 = A

y = B sin ω x 0 = B sin 0 = 0

At t = π 2 ω , the position of the particle is given by :

x = A cos ω x π / 2 ω = A cos π / 2 = 0

y = B sin ω x π / 2 ω = a sin π / 2 = B

Motion along an elliptical path

The linear distance equals displacement.

Therefore , the displacement in the given time interval is :

r = A 2 + B 2

Got questions? Get instant answers now!

Questions & Answers

A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
58asagravitasnal firce
Amar
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad
what is the definition of resolution of forces
Atinuke Reply
what is energy?
James Reply
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
أوك
عباس
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
Rama Reply

Get the best Physics for k-12 course in your pocket!





Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics for k-12' conversation and receive update notifications?

Ask