Question

A ball is thrown from the ground with nonzero horizontal and vertical initial velocities. While the ball is in the air, assuming that only the force of gravity acts on it, which of the following statements is true? a. Both the horizontal and vertical components of momentum are constant. b. Only the horizontal component of momentum is constant. c. Neither the horizontal nor the vertical components of momentum are constant.

Answer 1

B

Step-by-step explanation:

Solution:-

- Take a coordinate system as follows:

x: Directed along the ground

y: Directed vertical to ground

- We will assume that the initial vertical and horizontal non-zeroes velocities are given as follows:

`v_x_i = v_o_x\\\\v_y_i = v_o_y\\`

- After a ball is thrown it continues a path of parabolic projectile. The motion of the ball can be analyzed in each coordinate system. We will assume that effects of air-resistance are negligible.

- Therefore, only gravity acts on the ball in the vertical direction. We can use kinematic equation of motions to determine the velocity of ball in either ( x-y ) direction at any instant of time ( t ).

- Use first kinematic equation of motion in both x and y directions.

`v_x_f = v_o_x + a_x*t\\\\v_y_f = v_o_y + a_y*t\\`

- The accelerations ( ax and ay ) in the direction of each axis are to be determined. We know that the gravity acceleration ( g ) acts in vertical direction or along y-axis ( ay ) and always directed downwards while velocity is directed up. Since, we neglected the effects of air-resistance there is no acceleration in the x-direction ( ax = 0 ) .

`v_x_f = v_o_x\\\\v_y_f = v_o_y - gt\\`

- We see that the horizontal velocity of the ball ( vxf ) at any point in time remains equal to the initial horizontal velocity; hence, it is constant throughout the journey.

- However, the velocity of the ball in vertical direction( vyf ) is changing for every unit of time ( t ) under the influence of gravitational acceleration. Hence, it is not constant throughout the journey