Question

An object is thrown upward at an angle θ above the ground, eventually returning to earth. (a) Is there any place along the trajectory where the velocity and acceleration are perpendicular? If so, where? at the initial position of motion at the highest point in the trajectory at the final position of motion velocity and acceleration are never perpendicular

Answer 1

Answer:

Yes, at the highest point in the trajectory velocity and acceleration are perpendicular.

Explanation:

Let $x-$axis be the ground and $y-$axis is the height above the ground. Let the initial velocity of the object is $u$ having angle $\theta$ with respect to ground and the acceleration due to gravity , $g$, is acting in vertically downward direction as shown in figure.

As gravitational force is acting in vertical direction, so, it will not change the horizontal velocity of the object. So, at any instant throughout the projectile motion the $x-$component of the initial velocity will remailns constant, i.e. $u_x=u \cos \theta$.

While, due gravitational force, $y-$component of the initial velocity will change. Initially, $u_y=u \sin \theta$ is in vertically upward direction and gravitational force is actiongn vertically downward direction, so, at first $u_y$ will decrease untill it reaches the highest point of the trajectory as shown. At the highest point the vertical component of the velocity $u_y=0$, so, there is only horizontal component of the velocity. i.e $u_x= u \cos\theta$ .

Now, the resultant velocity of the object at the highest point is $u \cos \theta$ which is in the horizontal direction while the acceleration , $g$ , (due to gravily) is actiog in vertically downward direction.

Hence, at the highest point in the trajectory velocity and acceleration are perpendicular.