How Do You Calculate the Y Component of Velocity for an Arrow Shot at an Angle?

[disque]

Homework Statement

An arrow is shot into the air at an angle of 59.4degrees above the horizontal with a speed of 20.9 m/s.

(a) What are the x and y components of the velocity of the arrow 3 s after it leaves the bowstring?

Homework Equations

v(o)sin(theta)t+1/2(a)(t^2)

The Attempt at a Solution

I know the x is just trig and ends up being 10.6 but i don't understand how to find the y component. I thought you just used the above equation and plugged 3 in for t and -9.8 for a since its going down. I did that and I got the wrong answer. Any help would be much appreciated. Thanks a lot

 

[LowlyPion]

Observe that the Vy (with positive up) is continually slowed by gravity.

V_t = Vo - g*t

or

V₃ = Vo*sinθ - g*3

As you noted you have the x component already, so ...

 

[nlightner]

.

I would first clarify the given information and assumptions for the problem. It is important to mention that the problem assumes that the arrow is shot in a vacuum, neglecting air resistance, and that the acceleration due to gravity is constant at -9.8 m/s^2.

Next, I would explain the concept of vector components and their relationship to the overall velocity vector. In this case, the initial velocity of the arrow can be broken down into its x and y components, with the x component representing the horizontal velocity and the y component representing the vertical velocity.

To find the y component of velocity at 3 seconds, we can use the given equation v(o)sin(theta)t+1/2(a)(t^2) with the following values:

- v(o) = 20.9 m/s (initial speed of the arrow)
- sin(theta) = sin(59.4) = 0.857 (since theta is given in degrees)
- t = 3 s (time after the arrow is shot)
- a = -9.8 m/s^2 (acceleration due to gravity)

Substituting these values into the equation, we get:

v(y) = (20.9 m/s)(0.857)(3 s) + 1/2(-9.8 m/s^2)(3 s)^2

v(y) = 17.93 m/s - 44.1 m/s^2

v(y) = -26.17 m/s

Therefore, the y component of the velocity of the arrow 3 seconds after it leaves the bowstring is -26.17 m/s, indicating that the arrow is moving downwards with a decreasing velocity due to the acceleration of gravity. It is important to note that the negative sign indicates the direction of the velocity is downwards, as opposed to the positive direction of the initial velocity.

In conclusion, the x component of the velocity remains constant at 10.6 m/s, while the y component decreases from 20.9 m/s to -26.17 m/s due to the effects of gravity.