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I am a high school student and a have a fundamental doubt. It is said that when force is applied perpendicular to velocity, there is no change in kinetic energy since there is no change in speed. For change in kinetic energy to happen, the resultant force should have a tangential component. My question is, if a particle moving horizontally with some speed $V_x$ and a force acts along $y$ axis to increase its speed vertically from $0$ to $V_y$. Isn't the total speed now $√(V^2_x + V^2_y)$ Thus the speed has increased and so should kinetic energy. Please explain where I am going wrong.

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The perpendicular component of a force will never change the speed. This is true.

What is happening in your case is at the instant that the force along the y-axis is applied, it will, for that short, differentially small time, only change the direction of velocity slightly upwards, but have no effect on the speed. However, if the force keeps acting upwards -- and now the object has a velocity component in the upwards y-direction -- the force is no longer perpendicular (there is a tangential component) and thereby the speed will change.

The force's direction would also need to change in such a way that it always remains perpendicular to the object's velocity. An example of this is uniform circular motion.

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  • $\begingroup$ Ok, I think I get it, but you said that in a differentially small time it will change the direction of the velocity slightly upwards, so won't the particle now have a vertical component of speed, or that is also too small to be neglected. Thank you for your answer. $\endgroup$
    – arnav
    Jun 22, 2022 at 5:41
  • $\begingroup$ Yes, it will have a vertical component of velocity. And it is because there is a vertical component of velocity that the velocity will no longer be perpendicular to the vertical force. This means a component of the force is acting tangentially to the velocity, and that's where you get a change in speed. $\endgroup$
    – user256872
    Jun 22, 2022 at 5:45
  • $\begingroup$ Ah sorry I meant to say, IF the force is removed after that small amount of time, did the force change the speed as it changed the direction. Bcs now the particle has a vertical component, there is no force now, so in that small time the direction changed but did the speed change too? Or is that too small $\endgroup$
    – arnav
    Jun 22, 2022 at 5:53
  • $\begingroup$ If the (finite) time for which the perpendicular force is applied is small enough, then the speed will not change, but the velocity direction will, and will be slightly headed upwards. @arnav $\endgroup$
    – user256872
    Jun 22, 2022 at 5:54
  • $\begingroup$ What you could also do is continuously change the force direction (ensuring it's always perpendicular to speed). If you do that with a constant force (in magnitude), you'd get uniform circular motion. $\endgroup$
    – user256872
    Jun 22, 2022 at 5:55
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It is said that when force is applied perpendicular to velocity, there is no change in kinetic energy since there is no change in speed.
Exactly what happens when there is uniform circular motion.
System => object undergoing uniform circular motion.
The only force which is acting is the force causing centripetal acceleration whose direction is at right angle to the direction of the velocity so no work is done on the object by the force causing the centripetal acceleration,

My question is, if a particle moving horizontally with some speed $V_{\rm x}$ and a force acts along $y$-axis . . . .
System => the projectile
No force in the horizontal direction, thus no change in the horizontal component of velocity, and a downward gravitational attractive force which does work on the projectile which in turn changes its downward component of velocity had hence its kinetic energy.

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