A cart is propelled over an xy plane with acceleration components ax = 5.1 m/s2 and ay = -2.9 m/s2. Its initial velocity has components v0x = 8.2 m/s and v0y = 11.2 m/s. In unit-vector notation, what is the velocity of the cart when it reaches its greatest y coordinate?
27
Yo who knows
To find the velocity of the cart when it reaches its greatest y coordinate, we need to find the components of the velocity at that point.
To start, let's find the time it takes for the cart to reach its greatest y coordinate. We can use the y-component of the initial velocity (v0y) and the y-component of the acceleration (ay) to solve for the time (t):
ay = (v_y - v0y) / t
Plugging in the values we know:
-2.9 m/s^2 = (v_y - 11.2 m/s) / t
Now, let's find the time (t):
t = (v_y - 11.2 m/s) / -2.9 m/s^2
Next, we need to find the x-component of the velocity when the cart reaches its greatest y coordinate. We can use the x-component of the initial velocity (v0x) and the x-component of the acceleration (ax) to find the displacement in the x-direction (Δx) at the time (t) we just calculated:
ax = Δx / t^2
Plugging in the values we know:
5.1 m/s^2 = Δx / (t^2)
We can rearrange this equation to solve for Δx:
Δx = 5.1 m/s^2 * (t^2)
Now we have the displacement in the x-direction (Δx). Let's find the y-component of the velocity (v_y) at this point. We know that the y-component of the initial velocity (v0y) is negative, and the y-component of the acceleration (ay) is negative as well, so the cart is slowing down in the y-direction.
We can find the y-component of the displacement (Δy) using the equation:
Δy = v0y * t + (1/2) * ay * t^2
Plugging in the values we know:
Δy = 11.2 m/s * t + (1/2) * (-2.9 m/s^2) * t^2
Finally, to find the y-component of the velocity (v_y), we divide the y-component of the displacement (Δy) by the time (t):
v_y = Δy / t
Now we have both the x-component and the y-component of the velocity when the cart reaches its greatest y coordinate. We can put these values in unit-vector notation as follows:
v = v_x * i + v_y * j
where i and j are the unit vectors for the x-direction and y-direction, respectively.