Answer:
To address questions e, f, and g, let's first establish the context. Assuming we are dealing with a typical projectile motion problem, here's how we can proceed:Part e: Sketch the graphs of (1) ( v_x ) vs. ( t ), (2) ( v_y ) vs. ( t ), and (3) ( v_z ) vs. ( t ).
Explanation:
For projectile motion with constant acceleration due to gravity:( v_x ) vs. ( t ):Since there is no acceleration in the ( x )-direction (assuming no air resistance), ( v_x ) remains constant.The graph of ( v_x ) vs. ( t ) is a horizontal line at the value of the initial horizontal velocity, ( v_{x0} ).( v_y ) vs. ( t ):In the ( y )-direction, the object experiences constant acceleration due to gravity, ( g ).The velocity ( v_y ) at any time ( t ) can be expressed as ( v_y = v_{y0} - gt ) (taking downward as positive for gravity).The graph of ( v_y ) vs. ( t ) is a straight line with a slope of (-g), starting from ( v_{y0} ).( v_z ) vs. ( t ):If the object moves in a plane (2D motion), ( v_z = 0 ) and doesn't change with time.The graph of ( v_z ) vs. ( t ) is a horizontal line at ( v_z = 0 ).If the motion includes a component in the ( z )-direction with initial velocity ( v_{z0} ) and no acceleration, then:The graph of ( v_z ) vs. ( t ) is a horizontal line at the value of ( v_{z0} ).Part f: What is the acceleration ( a ) of the object as a function of time ( t )?For projectile motion under gravity:In the ( x )-direction, there is no acceleration: ( a_x = 0 ).In the ( y )-direction, the acceleration is constant and equal to (-g): ( a_y = -g ).In the ( z )-direction (for 2D motion): ( a_z = 0 ).If the motion includes the ( z )-direction with no acceleration:The acceleration ( a ) vector can be written as: [ \vec{a} = (0, -g, 0) ]Part g: Completely and accurately describe the motion of the object in the ( x ), ( y ), and ( z ) directionsMotion in the ( x )-direction:The object moves with a constant velocity ( v_x = v_{x0} ).The position in the ( x )-direction as a function of time is ( x(t) = x_0 + v_{x0} t ).Motion in the ( y )-direction:The object is under the influence of gravity, causing a constant acceleration (-g).The velocity in the ( y )-direction is ( v_y(t) = v_{y0} - gt ).The position in the ( y )-direction as a function of time is ( y(t) = y_0 + v_{y0} t - \frac{1}{2}gt^2 ).Motion in the ( z )-direction:For 2D motion, there is no movement in the ( z )-direction: ( v_z = 0 ) and ( z = z_0 ).If there is an initial velocity ( v_{z0} ) and no acceleration in the ( z )-direction, the velocity remains constant: ( v_z(t) = v_{z0} ), and the position is ( z(t) = z_0 + v_{z0} t ).Summary of the object's motion:( x )-direction: Uniform linear motion with constant velocity ( v_{x0} ).( y )-direction: Uniformly accelerated motion due to gravity.( z )-direction: No motion for 2D motion, or uniform linear motion with constant velocity ( v_{z0} ) if considered in 3D.This describes the overall motion of the object in the ( x ), ( y ), and ( z ) directions comprehensively.
