At what times are the velocity and acceleration perpendicular?

[Daniel Luo]

Homework Statement

The coordinates of the particle is given by:

x(t)=(1.2 m/s)t
y(t)=15m-(0.5m/s²)t²

a) At what times is the particle's velocity perpendicular to its acceleration
b) At what times is the particle's speed instantaneously not changing (I get that the acceleration is zero).
c) At what times is the particle's position perpendicular to its velocity.
d) What is the particle's minimum distance from the origin.

Homework Equations

r=xi+yj (i and j are the corresponding unit vectors)
v_x=dx/dt
v_y=dy/dt
a=dv/dt.

The Attempt at a Solution

I tried to differentiate the particle position r with respect to time getting velocity, and then taking the derivative of the velocity with respect to time getting acceleration.
When I had the acceleration as function of time, I got: a=(-2*0.5 m/s²)j. But how can I then solve for t, when it is not in the equation? What have I done wrong? Please help me with all the questions.

 

[mfb]

I got: a=(-2*0.5 m/s2)j

j is the y-direction? Okay.
So now you can insert the requirement "acceleration is orthogonal to the velocity". If the acceleration points in y-direction, which direction do you need for the velocity?
This will lead to an equation with time inside.

 

[verty]

For part a, use the dot product. For part b, think about centripetal acceleration, the speed can be constant even though there is acceleration. The condition you want is, the velocity and acceleration are orthogonal (perpendicular with the addition that the zero vector is orthogonal to every vector). The rest I'll leave to you.

Got beat by mfb but I think both replies are useful.

 

[LCKurtz]

Homework Statement

The coordinates of the particle is given by:

x(t)=(1.2 m/s)t
y(t)=15m-(0.5m/s²)t²

a) At what times is the particle's velocity perpendicular to its acceleration
b) At what times is the particle's speed instantaneously not changing (I get that the acceleration is zero).


c) At what times is the particle's position perpendicular to its velocity.
d) What is the particle's minimum distance from the origin.

Homework Equations

r=xi+yj (i and j are the corresponding unit vectors)
v_x=dx/dt
v_y=dy/dt
a=dv/dt.

The Attempt at a Solution

I tried to differentiate the particle position r with respect to time getting velocity, and then taking the derivative of the velocity with respect to time getting acceleration.
When I had the acceleration as function of time, I got: a=(-2*0.5 m/s²)j. But how can I then solve for t, when it is not in the equation? What have I done wrong? Please help me with all the questions.

Are you unable to simplify -2*.05? Just asking..

So the object has constant, nonzero, acceleration. There is no time t where the acceleration is zero, so that's why you can't make it work.

If you want help on the other parts, you need so show your work and explain where you are stuck.

 

[Ray Vickson]

Homework Statement

The coordinates of the particle is given by:

x(t)=(1.2 m/s)t
y(t)=15m-(0.5m/s²)t²

a) At what times is the particle's velocity perpendicular to its acceleration
b) At what times is the particle's speed instantaneously not changing (I get that the acceleration is zero).
c) At what times is the particle's position perpendicular to its velocity.
d) What is the particle's minimum distance from the origin.

Homework Equations

r=xi+yj (i and j are the corresponding unit vectors)
v_x=dx/dt
v_y=dy/dt
a=dv/dt.

The Attempt at a Solution

I tried to differentiate the particle position r with respect to time getting velocity, and then taking the derivative of the velocity with respect to time getting acceleration.
When I had the acceleration as function of time, I got: a=(-2*0.5 m/s²)j. But how can I then solve for t, when it is not in the equation? What have I done wrong? Please help me with all the questions.

Never, never include units like that in your equations; just write
x(t) = 1.2*t and y(t) = 15 -.5*t² (in m/sec). For example, if you tried to submit your equations---exactly as written---to a computer algebra package, the computer would choke, or else would severely misinterpret your formulas.

 

[verty]

Never, never include units like that in your equations; just write
x(t) = 1.2*t and y(t) = 15 -.5*t² (in m/sec). For example, if you tried to submit your equations---exactly as written---to a computer algebra package, the computer would choke, or else would severely misinterpret your formulas.

Some books teach that as a way to catch errors, it seems. Mistakes in calculations are pernicious, so it may be a useful investment of time in tests where one is usually under stress anyway.