<< Chapter < Page Chapter >> Page >

v t = d x d t đ x = v t đ t Δ x = v t đ t

This relation yields an expression of position in "t" after using initial conditions of motion.

Problem : The acceleration of a particle along x-axis varies with time as :

a = t - 1

Velocity and position both are zero at t= 0. Find displacement of the particle in the interval from t=1 to t = 3 s. Consider all measurements in SI unit.

Solution : Using integration result obtained earlier for expression of acceleration :

Δ v = v 2 - v 1 = a t đ t

Here, v 1 = 0 . Let v 2 = v , then :

v = 0 t a t đ t = 0 t t - 1 đ t

Note that we have integrated RHS between time 0 ans t seconds in order to get an expression of velocity in “t”.

v = t 2 2 t

Using integration result obtained earlier for expression of velocity : Δ x = v t đ t = t 2 2 t d t

Integrating between t=1 and t=3, we have :

Δ x = 1 3 t 2 2 t đ t = [ t 3 6 t 2 2 ] 1 3 = 27 6 9 2 1 6 + 1 2 = 5 4 = 1 m

Got questions? Get instant answers now!

Problem : If the velocity of particle moving along a straight line is proportional to 3 4 th power of time, then how do its displacement and acceleration vary with time?

Solution : Here, we need to find a higher order attribute (acceleration) and a lower order attribute displacement. We can find high order attribute by differentiation, whereas we can find lower order attribute by integration.

v t 3 4

Let,

v = k t 3 4

where k is a constant. The acceleration of the particle is :

a = đ v ( t ) đ t = 3 k t 3 4 - 1 4 = 3 k t - 1 4 4

Hence,

v t - 1 4

For lower order attribute “displacement”, we integrate the function :

đ x = v đ t Δ x = v đ t = k t 3 4 đ t = 4 k t 7 4 7 Δ x t 7 4

Got questions? Get instant answers now!

Velocity and acceleration is expressed in terms of time “x”

Let the expression of acceleration in x given as function a(x). Now, the defining equation of instantaneous acceleration is:

a x = đ v đ t

In order to incorporate differentiation with position, “x”, we rearrange the equation as :

a x = đ v đ x X đ x đ t = v đ v đ x

We obtain expression for velocity by rearranging and integrating :

v d v = a x d x

This relation yields an expression of velocity in x. We obtain expression for position/ displacement by using defining equation, rearranging and integrating :

v x = đ x đ t đ t = đ x v x Δ t = đ x v x

This relation yields an expression of position in t.

Problem : The acceleration of a particle along x-axis varies with position as :

a = 9 x

Velocity is zero at t = 0 and x=2m. Find speed at x=4. Consider all measurements in SI unit.

Solution : Using integration,

v đ v = a x đ x 0 v v đ v = 2 x 9 x đ x [ v 2 2 ] 0 v = 9 [ x 2 2 ] 2 x v 2 2 = 9 x 2 2 36 2 v 2 = 9 x 2 36 = 9 x 2 4 v = ± 3 x 2 4

We neglect negative sign as particle is moving in x-direction with positive acceleration. At x =4 m,

v = 3 4 2 4 = 3 12 = 6 3 m / s

Got questions? Get instant answers now!

Acceleration in terms of velocity

Let acceleration is expressed as function of velocity as :

a = a v

Obtaining expression of velocity in time, “t”

The acceleration is defined as :

Arranging terms with same variable on one side of the equation, we have :

đ t = đ v a v

Integrating, we have :

Δ t = đ v a v

Evaluation of this relation results an expression of velocity in t. Clearly, we can proceed as before to obtain expression of position in t.

Obtaining expression of velocity in time, “x”

In order to incorporate differentiation with position, “x”, we rearrange the defining equation of acceleration as :

a v = đ v đ x X đ x đ t = v đ v đ x

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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