Vector Composition & Decomposition Practice Problems

A trolley loaded with water bottles is pushed up an incline, making an angle of 15.0° above the horizontal. The force applied has a magnitude of 120 N and makes an angle of 15.0° counterclockwise with respect to the incline. Calculate the F_x (horizontal) and F_y (vertical) components of the applied force.

A ski jumper has an acceleration of magnitude 8.00 m/s² just before he leaves the ramp. The ramp makes an angle of 18.0° counterclockwise with respect to the horizontal. Calculate the a_x and a_y components of the skier's acceleration before he leaves the ramp.

A box is dragged on a ramp with a force of 8.0 N parallel to the surface. The ramp makes an angle θ above the horizontal.  The horizontal component of the force vector is 6.0 N. Calculate the vertical component of the force vector.

A car has a position vector of 3.0 m with a direction of 45° clockwise from the north. Determine the components of the position vector along the x and y axes in the normal and tilted coordinate systems shown below.

 Determine the components of the force F (25.0 N, north) along the x and y axes of the tilted coordinate system shown below.

Use the following component pairs to determine the magnitude and direction of vector given by: (a) M_x =  4.3  mm, M_y = -8.6 mm (b) N_X =  -4.5 m, N_y = -8.7 m

You are provided with two vectors D and E. Vector D lies in the first quadrant, with a length of 4.20 cm and 40.0 degrees above the x-axis. Vector E lies in the fourth quadrant, with a length of 2.40 cm and 40.0 degrees below the x-axis. Find the magnitude and direction of D - E  using components.

Sketch the acceleration vector (a) and calculate the a_x and a_y components of a chart pulled by a horse and accelerating at 2.00 m/s², 35.0° clockwise from the positive x-axis.

Using a convention where θ is a counterclockwise angle measured from the +x-axis to a vector M, determine the angle θ for a vector with the following components: (a) Mx = 3.00 m, My = -1.00 m; b) Nx = -5.00 m, Ny = -3.00 m.

You know that a particular vector M has a y-component M_y = +8.5 m and a counterclockwise angle of 48° from the +x-axis. What is the (a) x-component of M and (b) magnitude of M?

Use vector decomposition skills to determine the x- and y-components of the vector N.

A volleyball setter passes the ball to the outside hitter with a speed of 15.0 m/s. The velocity vector of the ball makes an angle of 20.0° with the vertical. Calculate the ball's horizontal velocity.

Sketch the velocity vector of a bicycle moving with a speed of 15.0 m/s in the direction of 20.0° east of the positive y-axis. Calculate the velocity vector v_x and v_y components.

Sketch the velocity vector (v) and calculate the v_x and v_y components of a hiker moving at 1.5 m/s, 25° counterclockwise from the negative x-axis.

R₁ (2.75 cm, 28 ° north of the negative x-axis) and R₂ (1.75 cm, 15 ° west of the positive y-axis) represent the position vectors of two charged particles. Calculate the x and y components of R₁ and R₂.

A child lets a toy car slide on the handrail of his house. The handrail is 40° inclined with respect to the horizontal. At the end of the handrail, the car has a speed of 4.5 m/s. Calculate the x and y components of the velocity vector at the instant when the car goes off of the handrail.

A cyclist climbs a steep road at a speed of 15.0 km/h. If the horizontal component of the velocity vector is 13.6 km/h, calculate the slope of the road.

A fridge is being pushed by a worker on the horizontal floor of an appliance store with a force (F) of magnitude 85.0 N and making an angle of 35.0° with respect to the vertical direction. Calculate the F_x and F_y components of the applied force.

A mountain climber of weight 637 N uses a fixed rope to ascend the inclined part of an artificial wall. The wall forms a right triangle structure. The wall's summit is 35.0 m above the ground and its base of 4.30 m is in contact with the ground. Calculate the magnitude of the climber's weight projection along the directions parallel and perpendicular to the inclined part of the wall.

A cyclist is accelerating up a 25.0° incline with an acceleration of 2.50 m/s². Determine the horizontal component of their acceleration.

A boat is moving at a velocity of 17.3 m/s in a direction 33.9° south of east as shown in the figure below. Calculate the southward and eastward components of the velocity vector.

A car experiences an acceleration $\overrightarrow{a}$ = (2.5î+1.8ĵ) m/s² in the xy-plane starting from a stationary position at t = 0. Find the horizontal and vertical components of velocity.

Consider a velocity vector in the xy-plane with a magnitude of 12 m/s. The x-component of the vector's velocity is -7.0 m/s. i) Find the two possible values for its y-component. ii) Assuming the y-component velocity is confirmed as positive, identify the extra velocity vector needed to alter the vector's direction to be along the positive y-axis and a magnitude of 20 m/s.

Consider a scenario where two forces, denoted as $\overrightarrow{F_1}$ and $\overrightarrow{F_2}$, are applied to an object resulting in a combined force $\overrightarrow{F_{r}}$ = $\overrightarrow{F_1}$ + $\overrightarrow{F_2}$. Describe the forces $\overrightarrow{F_1}$ and $\overrightarrow{F_2}$, if the magnitude of the resultant force squared, (F_r²), equals the sum of the squares of the magnitudes of the individual forces, (F₁² + F₂²).