Question:

A grasshopper hops along a level road. On each hop, the grasshopper launches itself at angle $\theta_0 = 45^\circ$ and achieves a range $R = 0.80 \textrm{ m}$. What is the average horizontal speed of the grasshopper as it hops along the road? Assume that the time spent on the ground between hops is negligible.

Source: Giancoli, Douglas C., Physics: Principles with Applications, 7th Edition, 2014.

Quick Answer:

$2.0\textrm{ m/s}$

### Transcript for this Giancoli solution

This is Giancoli Answers with Mr. Dychko. We'll use the range formula to figure out the initial velocity of the grasshopper and then knowing its angle is 45 degrees, figure out the*x*-component of its velocity and there we go. So we rearrange this to solve for

*v i*squared and we'll times by

*g*over

*sin*of 2

*Θ*both sides and then we get

*v*initial squared equals

*Rg*over

*sin 2Θ*. Take the square root of both sides and we solve for the initial velocity and so that will be square root of 0.8 meters times 9.8 meters per second squared divided by sin of 90 because that's 2 times 45 and we get 2.8 meters per second. And then the horizontal component of its velocity is the adjacent leg of this right triangle here and so we use cosine 45 to figure that out. And we multiply 2.8 times

*cos 45*to get 2.0 meters per second is the horizontal velocity that's two significant figures since we had two significant figures in our range.

## Comments

Hi! Would you be able to explain something to me? I don't understand why solving for that x component gives the average horizontal speed. I thought that x component is only applicable to the instantaneous initial velocity?

Hi sophiaswimgirl,

Thanks a lot for the question, and I'm sorry for taking so long to get back. I hope you're still working on this unit....

Sure, it's true that the x-component of the velocity is that of the instantaneous initial velocity, as you say. However, the x-component of the velocity never changes. There is no horizontal acceleration. Since the x-component of the velocity never changes, this means that whatever value it has initially will also be the value it has at any other time, and so in this circumstance it's initial value is also it's average. The average of a value that's constant is whatever that value is, at any time.

Hope this helps,

Mr. Dychko