"speed rate of change" vs. "velocity rate of change"

In summary, the conversation discusses finding the rate of change of the speed of a particle at a given moment in time. The particle's velocity and acceleration are given, and the equation a=dv/dt is used to solve for the rate of change of speed. The correct answer is 6 m/s2, and the conversation also includes a discussion about differentiating the equation |v|^2=√(x^2+y^2) to find this result.
  • #1
cj
85
0

Homework Statement


At a moment in time a particle has a velocity of v = 3 m/s î + 4 m/s ĵ, and an acceleration of a = 6 m/s2 î + 3 m/s2 ĵ. Find the rate of change of the speed of the particle, that is, find d|v|/dt.
(better formatted reference: http://content.screencast.com/users/cjwood99/folders/Default/media/5b55184b-8954-4fb0-9b89-588e52d22038/Capture.JPG)​

The answer is: 6 m/s2. How on Earth?​

Homework Equations


speed = magnitude of velocity
a = dv/dt​

The Attempt at a Solution


The magnitude of the initial velocity (the speed) is 5 m/s. 3^2 + 4^2 = 5^2.
Acceleration is constant, so I tried using the equations of uniform acceleration kinematics, e.g. vx=∫(ax)dt, etc.
Can't, for the life of me, see how the rate of change of speed = 6m/s2.
Thanks for any hints.​
 
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  • #2
cj said:

Homework Statement


At a moment in time a particle has a velocity of v = 3 m/s î + 4 m/s ĵ, and an acceleration of a = 6 m/s2 î + 3 m/s2 ĵ. Find the rate of change of the speed of the particle, that is, find d|v|/dt.
(better formatted reference: http://content.screencast.com/users/cjwood99/folders/Default/media/5b55184b-8954-4fb0-9b89-588e52d22038/Capture.JPG)​

The answer is: 6 m/s2. How on Earth?​

Homework Equations


speed = magnitude of velocity
a = dv/dt​

The Attempt at a Solution


The magnitude of the initial velocity (the speed) is 5 m/s. 3^2 + 4^2 = 5^2.
Acceleration is constant, so I tried using the equations of uniform acceleration kinematics, e.g. vx=∫(ax)dt, etc.
Can't, for the life of me, see how the rate of change of speed = 6m/s2.
Thanks for any hints.​

Write the velocity as function of time, determine the speed as function of time, then differentiate. That is the rate of change of speed
at time t. You need the rate of change at t=0.
 
  • #3
ehild said:
Write the velocity as function of time, determine the speed as function of time, then differentiate. That is the rate of change of speed
at time t. You need the rate of change at t=0.
Hmm... trying to follow your advice.
http://content.screencast.com/users/cjwood99/folders/Default/media/71db2391-5348-4a08-ae6c-d4035fc5f593/Picture1.jpg
The stated answer is 6 m/s2, which I'm not finding any way to get. Maybe 6 m/s2 is incorrect?
 
  • #4
cj said:
Hmm... trying to follow your advice.
http://content.screencast.com/users/cjwood99/folders/Default/media/71db2391-5348-4a08-ae6c-d4035fc5f593/Picture1.jpg
The stated answer is 6 m/s2, which I'm not finding any way to get. Maybe 6 m/s2 is incorrect?
The right hand side of the first line of step 6 is wrong. How do you differentiate ##\sqrt{x^2+y^2}##?
An easier way is to start with ##|v|^2=\vec v.\vec v## and differentiate both sides.
 
  • #5
haruspex said:
The right hand side of the first line of step 6 is wrong. How do you differentiate ##\sqrt{x^2+y^2}##?
An easier way is to start with ##|v|^2=\vec v.\vec v## and differentiate both sides.
Ah! Got it! Thank you so much.
 

What is the difference between speed rate of change and velocity rate of change?

The difference between speed rate of change and velocity rate of change lies in their respective definitions. Speed rate of change is defined as the rate at which an object's distance changes over time, regardless of the direction. On the other hand, velocity rate of change is defined as the rate at which an object's displacement changes over time in a specific direction. In simpler terms, speed rate of change is a scalar quantity while velocity rate of change is a vector quantity.

Can an object have a constant speed rate of change but varying velocity rate of change?

Yes, it is possible for an object to have a constant speed rate of change but varying velocity rate of change. This can occur when an object moves along a curved path, as its velocity is constantly changing in direction even though its speed remains constant. This is because velocity takes into account the object's direction of motion, while speed does not.

How is the speed rate of change related to the velocity rate of change?

The speed rate of change and the velocity rate of change are related through a mathematical formula. The velocity rate of change is equal to the speed rate of change multiplied by the direction of motion. This means that the speed rate of change can be considered as the magnitude of the velocity rate of change.

Which is more important in determining an object's motion: speed rate of change or velocity rate of change?

Both the speed rate of change and the velocity rate of change are important in determining an object's motion. The speed rate of change tells us how fast an object is moving, while the velocity rate of change tells us how the object is moving. In other words, speed rate of change gives us the magnitude of the motion, while velocity rate of change gives us the direction of the motion. Both quantities are necessary to fully understand an object's movement.

Can the speed rate of change and velocity rate of change be negative?

Yes, both speed rate of change and velocity rate of change can be negative. A negative speed rate of change would indicate that the object is moving backwards or in the opposite direction of its initial motion. A negative velocity rate of change would indicate that the object is moving in the opposite direction of its initial motion, regardless of its speed. In both cases, the negative sign represents a change in direction.

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