Motion in a Plane: Velocity & Acceleration

[Elena14]

A particle has initial velocity 3i + 4j and acceleration of 0.4i + 0.3j. what is its speed after 10 seconds?
I know the correct way to solve this question is by dealing with x and y components separately and it gets us to the answer 7√2. But what if we first find the magnitude of velocity and acceleration and then use v=u+at to find speed. This gets us to the answer 10.
I know that we deal with x and y components of any vector separately but when we are just dealing with magnitude of vector, the second method should as well be right.
Why are the answers different and where am I wrong?

 

[PeroK]

A particle has initial velocity 3i + 4j and acceleration of 0.4i + 0.3j. what is its speed after 10 seconds?
I know the correct way to solve this question is by dealing with x and y components separately and it gets us to the answer 7√2. But what if we first find the magnitude of velocity and acceleration and then use v=u+at to find speed. This gets us to the answer 10.
I know that we deal with x and y components of any vector separately but when we are just dealing with magnitude of vector, the second method should as well be right.
Why are the answers different and where am I wrong?

This problem shows you that can't use $v = u + at$ for vector magnitudes. You might like to experiment with different velocities and accelerations to see why it goes wrong. You may not yet have covered circular motion, but for circular motion at constant speed there is a constant acceleration (called centripetal) towards the centre. In that case, the speed doesn't change, despite the acceleration, which shows that $v = u + at$ cannot be true.

 

[Elena14]

Why is it that the equation is not applicable for "vector magnitudes"?

 

[PeroK]

Why is it that the equation is not applicable for "vector magnitudes"?

You should try some simple examples yourself. E.g. try with initial velocity in the x direction and acceleration in the y direction. You will see for yourself.

 

[Elena14]

We can always verify it but what is the reason behind the equations not being applicable for vector magnitudes?

 

[cnh1995]

A particle has initial velocity 3i + 4j and acceleration of 0.4i + 0.3j.

These two represent both magnitude and direction of the vectors.
v=u+at would be true only if v,u and a are in the same direction.

 

[Elena14]

And what if we didn't know the language of vectors. Would we be never be able to understand motion in more than one dimension?

 

[cnh1995]

And what if we didn't know the language of vectors. Would we be never be able to understand motion in more than one dimension?

No.

 

[Elena14]

No.

I get this but can you please explain to me why the equations of motion are NOT applicable for vector magnitudes?

 

[PeroK]

I get this but can you please explain to me why the equations of motion are NOT applicable for vector magnitudes?

If you're not prepared to do any work yourself, why should we spend time doing it for you?

 

[cnh1995]

I get this but can you please explain to me why the equations of motion are NOT applicable for vector magnitudes?

I believe I tried it in #6..

 

[Elena14]

If you're not prepared to do any work yourself, why should we spend time doing it for you?

I did try to figure it out myself. I tried to go in the very derivation of the equations of motion (by calculus method) but that doesn't tell me why the equations should not be applicable. I do understand the fact that they are applicable only when velocity and acceleration are in the same direction. Everyone keeps giving this explanation without trying to point out why the equations don't work when the velocity and acceleration vectors are not in the same direction.

 

[cnh1995]

Everyone keeps giving this explanation without trying to point out why the equations don't work when the vector are not in the same direction.

They will work if you consider the angle between the two vectors and modify the equations accordingly. When you release a ball, acceleration due to gravity will make it fall vertically downward. Here, "total gravity" makes the ball fall, hence a=g. But if the ball rolls down an incline, its acceleration is gsinθ, since component of the gravitational force "along" the incline is responsible for the ball's motion. You need to consider the components of vectors which are responsible for a particular motion, when the vectors are not along the same line.

 

[PeroK]

I did try to figure it out myself. I tried to go in the very derivation of the equations of motion (by calculus method) but that doesn't tell me why the equations should not be applicable. I do understand the fact that they are applicable only when velocity and acceleration are in the same direction. Everyone keeps giving this explanation without trying to point out why the equations don't work when the velocity and acceleration vectors are not in the same direction.

Look at what I said in post #4. You can work it out for yourself with a bit of elementary geometry.

 

[brotherStefan]

We can always verify it but what is the reason behind the equations not being applicable for vector magnitudes?

Draw the vector (3i + 4j) on a piece of graph paper. When you measure the length (it's magnitude or absolute value) of this vector, you will find it has a length of 5 units, even though the sum of the i and j components is 7. Now draw the vector (3i - 4j) and measure the length of this vector. It, too, has a length of 5 units, but the simple sum of it's two components is -1. Even though the two vectors have the same magnitude, they are two completely different vectors -- the two vectors represent two different complex numbers. The computation of complex numbers requires the use of the rules for adding and/or multiplying complex numbers. The alternative is to do the computations graphically.