Question

A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t = 0 s, the xcomponents of the puck's initial velocity and acceleration are v_0x = +1.2 m/s and a_x = +6.6 m/s². The ycomponents of the puck's initial velocity and acceleration are v_0y = +6.5 m/s and a_y = -3.3 m/s². Find (a) the magnitude v and (b) the direction θ of the puck's velocity at a time of t = 0.50 s. Specify the direction relative to the +x axis

Answer 1

The x componet of the velocity after $t = 0.5s$ is:

$$v_x = v_{0x} + a_xt\\ v_x =1.2m/s + 6.6m/s^2\cdot 0.5 = 4.5m/s$$

The y componet of the velocity after $t = 0.5s$ is:

$$v_y = v_{0y} + a_yt\\ v_y =6.5m/s -3.3m/s^2\cdot 0.5 =4.85m/s$$

The magnitude of the velocity is then (by Phyphagorean's theorem):

$$v = \sqrt{v_{x}^2+v_{y}^2}\\ v = \sqrt{4.5^2+4.85^2} \approx 6.62m/s$$

The angle relative to the positive x axis is:

$$\theta = \arctan\left(\dfrac{v_y}{v_x} \right)\\ \theta = \arctan\left(\dfrac{4.85}{4.5} \right) \approx 47.1\degree$$

Answer. a) 6.62m/s, b) 47.1 degree.