Velocity Using Parametric Equations

In summary, to find the magnitude of the total velocity of an object moving in two dimensions, we use the parametric equations x(t) = At^2 + Bt and y(t) = D cos(Et), with constants A = 2 m/s^2, B = 3 m/s, D = 4 m, and E = 1 rad/s. To find the magnitude of the total velocity at t = 3 s, we plug in the values for t into the derivatives of x(t) and y(t), square and sum these values, and then take the square root of the sum. This gives us the magnitude of the vector \vec{v}=[v_x,v_y].
  • #1
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Homework Statement



An object moves in two dimensions according to the parametric equations x(t) = At^2 + Bt and y(t) = D cos(Et). The constants A, B, D, and E are A = 2 m/s^2, B = 3 m/s, D = 4 m, and E = 1 rad/s. What is the magnitude of the total velocity of the object at t = 3 s?

Homework Equations





The Attempt at a Solution



I'm not sure if I did this problem right. I plugged back in the constants

x(t) = 2 t^2 + 3 t
y(t) = 4 cos(t)

dx/dt = 4 t + 3
dy/dt = -4 sin(t)

dy/dx = dy/dt dt/dx = [-4 sin(t)]/[4 t + 3]

I thought that this was the velocity?

I then plugged in 3 for t and then plugged this into my calculator
[-4 sin(3)]/(12+3) and got about - .038 m/s but sense it said magnitude only I ignored the negative sign and put .038 m/s

I have the feeling I did this problem wrong. This is for my physics 2 course and is suppose to be a introductory physics course after taking physics 1 (non calculus based) and this is just suppose to be like calculus I based but parametric equations is a calculus 2 topic (in most american schools) and I'm in calculus 2 at the moment and haven't covered the topic yet and only have a brief understanding of it so I'm not sure
 
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  • #2
What is the magnitude of the vector [itex]\vec{v}=[v_x,v_y][/itex]?
Or, how do you evaluate the magnitude of a vector knowing its components?

And, [itex]\vec{v}=[v_x,v_y]=[\frac{dx}{dt},\frac{dy}{dt}][/itex]
 
  • #3
Ah I thought so, so I plug in 3 into each of the derivatives square both of these values sum these squared values and then take the square root of the whole thing?
 
  • #4
Yep
[itex]|\vec{v}_{t=3}|=\sqrt{\left[\left.\frac{dx}{dt}\right|_{t=3}\right]^2+\left[\left.\frac{dy}{dt}\right|_{t=3}\right]^2}[/itex]
 
  • #5
if I did this problem right.

I would first verify that the parametric equations given are correct and accurately represent the motion of the object. I would also check that the units of the constants A, B, D, and E are consistent with the given units for velocity. Assuming the equations are correct, I would proceed to calculate the total velocity at t=3s.

To do this, I would first calculate the individual components of velocity, dx/dt and dy/dt, at t=3s using the given parametric equations. From there, I would use the Pythagorean theorem to find the magnitude of the total velocity, which is the square root of the sum of the squares of the individual components of velocity.

In this case, the individual components of velocity at t=3s are dx/dt = 15 m/s and dy/dt = -3.43 m/s. Plugging these values into the Pythagorean theorem, we get the magnitude of the total velocity to be approximately 15.6 m/s.

It is important to note that the negative sign for dy/dt indicates the direction of the velocity, which would be downwards in this case. So the total velocity at t=3s would have a magnitude of 15.6 m/s and a downward direction.
 

1. What are parametric equations?

Parametric equations are a set of equations that describe the relationship between two or more variables in terms of a third variable, known as the parameter. They are commonly used in mathematics and physics to represent curves and motion.

2. How is velocity calculated using parametric equations?

To calculate velocity using parametric equations, we first find the derivative of the parametric equations with respect to the parameter. This derivative represents the rate of change of the position of an object with respect to time, which is the definition of velocity.

3. What is the difference between velocity and speed?

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time, including its direction. Speed, on the other hand, is a scalar quantity that only describes the rate of change of an object's position without considering its direction.

4. How do parametric equations help in understanding motion?

Parametric equations help in understanding motion by providing a mathematical representation of the position, velocity, and acceleration of an object at any given time. This allows us to analyze the motion and make predictions about the future behavior of the object.

5. Can parametric equations be used to describe any type of motion?

Yes, parametric equations can be used to describe any type of motion as long as it can be represented by a function of time. This includes linear, circular, and projectile motion, among others.

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