Vector and acceleration problem

[Zeitgeist]

Homework Statement

A bird takes 8.5s to fly from position A{ V = 4.4 m/s [31° S of E]} to position B{ v = 7.8 m/s [25° N of E]}. Determine the average acceleration.

Homework Equations

$$a_{ave} = \Delta V / \Delta t$$
$$\Delta Vx = Vbx + (-Vax)$$
$$\Delta Vy = Vby + (-Vay)$$

The Attempt at a Solution

$$\Delta Vx = Vbx + (-Vax)$$
= $$Vb \sin \Theta + (-Va \cos \beta)$$
= $$7.8 m/s (\sin31) - 4.4 m/s(\cos25)$$
= 0.0295 m/s

$$\Delta Vy = Vby + (-Vax)$$
= $$Vb \cos \Theta + (-Va \sin \beta)$$
= $$7.8 m/s (\cos31) + 4.4 m/s (\sin25)$$
= 8.545 m/s

$$\mid \Delta V\mid^2 = \mid \Delta Vx \mid^2 + \mid \Delta Vy \mid^2$$
= $$(0.0295)^2 + (8.545)^2$$
= 8.545 m/s

$$a_{ave} = \Delta V / \Delta t$$
= (8.545m/s) / (8.5s)
= 1.005

This is how far I've got and I checked the answer with my textbook it says that the answer should be 0.76 m/s ^2.

Can someone help please?

 

[anathema]

There are a few issues with your solution. First, you have calculated the magnitude of the change in velocity, but you need to find the components of the change in velocity in the x and y directions separately. This can be done using trigonometry and the given angles.

Next, you have used the wrong formula for average acceleration. The correct formula is a_{ave} = \Delta V / \Delta t, where \Delta V is the change in velocity and \Delta t is the time interval. In your calculation, you have used the magnitude of the change in velocity instead of the actual change in velocity.

Finally, when you calculate the average acceleration, you should use the components of the change in velocity that you calculated earlier, not the magnitude.

Here is the correct solution:

\Delta Vx = Vbx - Vax = 7.8 m/s \sin 25^{\circ} - 4.4 m/s \sin 31^{\circ} = 2.02 m/s

\Delta Vy = Vby - Vay = 7.8 m/s \cos 25^{\circ} - 4.4 m/s \cos 31^{\circ} = 0.96 m/s

a_{ave} = \Delta V / \Delta t = (2.02 m/s \hat{i} + 0.96 m/s \hat{j}) / 8.5 s = 0.76 m/s^2

Therefore, the average acceleration is 0.76 m/s^2.

 

[m2003]

It looks like you have made a mistake in your calculation for the magnitude of the change in velocity. The correct equation for this is:

|ΔV| = √(ΔVx^2 + ΔVy^2)

Using this, the magnitude of the change in velocity is:

|ΔV| = √(0.0295^2 + 8.545^2) = 8.546 m/s

And the average acceleration is:

a_{ave} = |ΔV| / Δt = (8.546 m/s) / (8.5 s) = 1.005 m/s^2

This matches your initial calculation, so it seems that the answer in the textbook is incorrect. I would recommend double-checking the solution or consulting with your teacher or classmates to confirm the correct answer.