A particle is moving in the xy plane with constant acceleration. At t=o, the particle is given by x=4m, y=3m, and has a velocity v=2m/s x-hat - 9m/s y-hat. The acceleration of the particle is given by a=4m/s^2 x-hat + 3m/s^2 y-hat. Find the velocity vector at t=2 seconds.

Question

A particle is moving in the xy plane with constant acceleration. At t=o, the particle is given by x=4m, y=3m, and has a velocity v=2m/s x-hat - 9m/s y-hat. The acceleration of the particle is given by a=4m/s^2 x-hat + 3m/s^2 y-hat. Find the velocity vector at t=2 seconds.

My only problem is i don't know which equation to use. I thought it would be V=Vo+a*t but I need to use the x component (displacement) & if i use V^2=Vo^2+2a*deltaX then I need a component for time which this equation doesnt have.

Answer 1

To find the velocity vector at time t=2 seconds, we can use the equations of motion for constant acceleration in two dimensions. The equation V = Vo + a*t is indeed applicable here, but we need to consider the vector components separately.

Let's break down the problem step by step:

Step 1: Determine the initial velocity components.
Given: v = 2 m/s x-hat - 9 m/s y-hat
The x-component of the initial velocity (Vo_x) is 2 m/s, and the y-component of the initial velocity (Vo_y) is -9 m/s.

Step 2: Determine the displacement components.
Given: x = 4 m and y = 3 m
Since the particle is moving with constant acceleration, we can use the equations of motion to find the displacement components at time t=2 seconds.
For the x-component of displacement (deltaX), we have: deltaX = Vo_x*t + 0.5*a_x*t^2.
Substituting the given values, deltaX = (2 m/s)*(2 s) + 0.5*(4 m/s^2)*(2 s)^2 = 4 m + 8 m = 12 m.
For the y-component of displacement (deltaY), we have: deltaY = Vo_y*t + 0.5*a_y*t^2.
Substituting the given values, deltaY = (-9 m/s)*(2 s) + 0.5*(3 m/s^2)*(2 s)^2 = -18 m + 6 m = -12 m.

Step 3: Determine the final velocity components.
Using the equation V = Vo + a*t, we can find the final velocity components.
For the x-component of velocity (V_x), we have: V_x = Vo_x + a_x*t.
Substituting the given values, V_x = (2 m/s) + (4 m/s^2)*(2 s) = 2 m/s + 8 m/s = 10 m/s.
For the y-component of velocity (V_y), we have: V_y = Vo_y + a_y*t.
Substituting the given values, V_y = (-9 m/s) + (3 m/s^2)*(2 s) = -9 m/s + 6 m/s = -3 m/s.

Step 4: Combine the components to find the velocity vector.
The velocity vector V at t=2 seconds is given by V = V_x x-hat + V_y y-hat
Substituting the final velocity components, we have: V = 10 m/s x-hat - 3 m/s y-hat.

Therefore, the velocity vector at t=2 seconds is 10 m/s x-hat - 3 m/s y-hat.