Speed of a charged particle?

Question

Hi-
Can someone please answer this for me? How do you show that if the force of an object is always perpendicular to the velocity of the object, then the speed of the object is constant?
Any help would be appreciated!!
Monica

Answer 1

First, we observe that an object's acceleration, a, is always in the same direction as the net force. Therefore, the acceleration is also perpendicular to the velocity, v. That is, letting * denote the dot product, we know v * a = 0.

Instead of showing that speed, s, is constant, it is easier to show that v * v = s^2 is constant. To do this, start by taking the derivative of v * v, using the rules of vector calculus:

d/dt ( v * v ) = (dv / dt) * v + v * (dv / dt)

Since dv/dt is nothing more than the acceleration vector a we get:

d/dt ( v * v ) = a * v + v * a

By symmetry of the dot product this is just:

d/dt ( v * v ) = 2(v * a)

Since v * a = 0 we get:

d/dt ( v * v ) = 0.

This means v * v must be constant. Since v * v is the square of the speed, we conclude that the speed must also be constant. QED

NOTE: This does *not* say that the velocity is constant. This would be impossible, since the object is being accelerated. Only the *magnitude* of the velocity is constant.

Answer 2

When the particle is travelling in constant straight path and force is applied perpendicular to it, then it goes in a circular path. Kind of like a satelite. The moon has a costant velocity due to which it has a tendency to move away from earth. But the earhts gravitational pull acts perpendicular to this velocity and thus moon has circular path. Since the force is constant thus it follows a circular path and not a spiral path. Thus the speed of of your object always remains constant.

Answer 3

In the elliptical orbital motion of the Earth revolving around the
Sun rotational energy is involved in maintaining rotational equilibrium .Since the radius of the orbit continulaly changes we have changing velocities and acceleration around the loop.The acceleration of the earth relative to the fulcrum of sun-earth,is a vector which on the ecliptic has two components; the tangential (is at right angle to the centripetal force)acceleration and the
radial acceleration wich is in the direction of the centrepetal force.
The Total acceleration is the vector sum of the two components.
Since the accelerationof the earth is a changing vector with rotational time ,it follows that the velocity of the earth around the loop is not constant.

Answer 4

If there is a non-zero net force acting on an object then the speed is definitely not constant. Remember Newton's first law.

Answer 5

Throw a punch in the air... Throw a ball across the room... There is the answer to your question

Answer 6

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