How do you prove?

Question

If the speed of an object is constant, then its velocity vector and acceleration vectors must be perpendicular.

How do you prove this?

Answer 1

Speed is a scalar. It is an absolute described as distance / time.

Add in a direction, and you'll get a vector. Velocity. Speed + distance = Velocity.

Since you say that it is traveling at constant speed, then distance divided by time must be constant.

Let's say, a car going 100 miles / 4 hrs = 25mph.

Now, let's make that speed into a velocity.

25 mph, heading exactly west.

Acceleration is change in velocity over time.

If we went faster, say 50 mph, then we would be accelerating since velocity would change.

Since the speed is constant (25 mph) if we headed a different direction, we'd also be accelerating since speed + direction is velocity.

Ultimately, we have to have a situation where we will a) Not change speed, and b) Not change direction. The only way to do this is to have the acceleration vector be perpendicular.

Imagine the car heading at 25 mph as an arrow heading west.

------------------>

If we take an acceleration vector (another large car which will smash ours and push it in another direction) how can we add in this second arrow to keep our car going 25 mph?

---------------> + ????


We cannot put it behind our car since the extra speed would speed it up.

-----------> + ------------> = ---------------------------------->

We can't smash from the front since this would slow it down.

If we contact from any other direction, we will change it's direction and hence accelerate it.

Therefore, the vector is perpendicular and has magnitude of zero.

You can probably find the mathematical proof online. I tried to use a tangible example.

Hope it helped.

Regards,

Mysstere

Answer 2

Speed is the magnitude of velocity. Acceleration is the rate of change in velocity.

If your acceleration vector is "a" m/s^2 [direction], that means that every second, the vector "a" m/s [direction] is added to your velocity vector to give you your new velocity.

Now, the vector "a" m/s [direction] has a component in the direction of your velocity and a component in the direction perpendicular to your velocity. When you add this vector to your velocity vector, the magnitude of your velocity will increase by the magnitude of the component of the acceleration vector in the direction of the velocity.

However, the magnitude of the velocity is your speed, and your speed must stay the same. Therefore, the component of the acceleration vector in the direction of your velocity vector must be 0. Therefore, your acceleration vector can only have a component in the direction perpendicular to your velocity vector.

Therefore, your acceleration vector will be perpendicular to your velocity vector.

Answer 3

Speed is the magnitude of the velocity vector; s=|v|. Then the dot product
v.v=|v|^2
is constant.
Now take a derivative:
0=2v.a,
so v and a are perpendicular.

Answer 4

it is circular motion
if speed is constant and velocity is changing only the direction should be changing
and when that is so the acceleration should be directed towards the centre and the direction of the velocity being that of the tangent of the circle thus mutually perpendicular

Answer 5

let's assume that the v^ is the velocity vector, a^ is the acceleration vector

v^(at time t)= v^(at time 0)+ a^*t

if we have a two coordinates system, and we analyzed the velocity scalar at x direction, then we have:

|v|cos(l) at x, t = |v|cos(M) at (t=0) + |a|cos(B)

as l,M,B are angles
ignoring (For some reason), the angle change in velocity
and saying that v is in x direction, then

v= v (at time zero) + a* cos B

if B=90 degrees, y-axis, then

v=v (at time zero)

speed is constant