What is the angle between velocity and acceleration at the highest point of a projectile motion

CONCEPT:

Projectile motion

If an object is given an initial velocity in any direction and then allowed to travel freely under gravity, then the object is called a projectile and the motion of the projectile is called the projectile motion.
There is no force other than the gravity acts on the projectile during the flight.

EXPLANATION:

When an object is projected at an angle from the surface of the earth, then there will be two components of the velocity.
One component of velocity will act along the horizontal direction and the other will act in the vertical direction.
There is no force other than the gravity acts on the projectile during the flight. So only one acceleration due to gravity in the vertically downward direction will act on the object.
Since there is no acceleration in the horizontal direction so the horizontal velocity will remain constant during the whole flight.
At the highest point, the vertical velocity of the object will be zero and the whole velocity of the particle will be along the horizontal direction.
The acceleration on the particle during the flight will be due to gravity, so the direction of acceleration will be vertical downward at the highest point.
So at the highest point, the direction of velocity is horizontal and the direction of acceleration is vertically downward, so the angle between the direction of velocity and acceleration at the highest point of a projectile path is 90°. Hence, option 3 is correct.

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Suppose I have a projectile and launch it with an oblique velocity, meaning it has both horizontal and vertical components of velocity. Neglecting air resistance and the like, it’s trajectory should look like what I drew-

Here the red arrow is the downward constant acceleration, the green arrow is the net velocity (vector sum of its components) and the angle between the acceleration and velocity I have marked in yellow.

I have filled in the angle in the regions where I think the angle between the velocity and the acceleration would be acute, which I think is the entirety of the projectile motion after it crosses its maximum height till it reaches the ground again.

However, on searching the net and also from a question paper I saw, it states the the angle between velocity and acceleration is minimum and acute at only one point.

Why is it only one point? Does the ‘minimum’ word have anything to do with it? I thought the angle would be acute over a number of points in the second half of the projectile’s motion.

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Text Solution

`0^(@)` `45^(@)``90^(@)` `180^(@)`

Answer : C

Solution : At the highest point, velocity is acting horizontally and acceleration ( = acceleration due to gravity) is acting vertically downwards. Therefore, at the highest point the angle between velocity and acceleration is `90^(@)` .