A small toy airplane is flying in the xy-plane parallel to the ground. In the time interval t=0 to t=10.0 s, its velocity as a function of time is given by υ⃗ =(1.50m/s2)ti^+[13.0m/s−(2.00m/s2)t]j^.
To find the displacement of the toy airplane during the time interval t=0 to t=10.0 s, we need to integrate the given velocity vector function.
The velocity vector function is given by υ⃗ =(1.50m/s2)ti^+[13.0m/s−(2.00m/s2)t]j^.
To integrate, we can break down the velocity vector function into its x and y components:
The x-component of the velocity vector is given by υx = (1.50 m/s^2)t, and
The y-component of the velocity vector is given by υy = [13.0 m/s - (2.00 m/s^2)t].
Now, let's integrate each component separately:
For the x-component:
The displacement in the x-direction (Δx) is given by integrating the x-component of the velocity with respect to time, i.e., Δx = ∫(1.50 m/s^2)t dt
Integrating this, we get Δx = (1.50/2) t^2 + Cx, where Cx is the constant of integration.
To find the value of Cx, we need the initial condition. The problem does not provide the initial condition, so we assume the airplane starts from the origin at t=0 s. Therefore, Cx = 0.
So, the displacement in the x-direction is given by Δx = (1.50/2) t^2.
For the y-component:
The displacement in the y-direction (Δy) is given by integrating the y-component of the velocity with respect to time, i.e., Δy = ∫[13.0 m/s - (2.00 m/s^2)t] dt
Integrating this, we get Δy = (13.0 t - (2.00/2) t^2) + Cy, where Cy is the constant of integration.
To find the value of Cy, we again assume the airplane starts from the origin at t=0 s. Therefore, Cy = 0.
So, the displacement in the y-direction is given by Δy = 13.0 t - (2.00/2) t^2.
Therefore, the total displacement of the toy airplane during the time interval t=0 to t=10.0 s is given by the vector Δ⃗ = Δx i^ + Δy j^.
Substituting the respective equations, we get:
Δ⃗ = (1.50/2) t^2 i^ + (13.0 t - (2.00/2) t^2) j^.
You can now substitute the value of t (between 0 and 10.0 s) into the above equation to find the displacement vector Δ⃗.