Will the Baseball Clear the Fence?

[dani123]

1. Homework Statement

A baseball is hit at a point 1.00m above homeplate. The baseball is given an initial velocity of 35.0m/s at an angle of 45.0° with respect to the horizontal. If the baseball is traveling towards the outfield fence which is 4.0m high and 120m away, will the baseball clear (go over) the fence? If so, by how many maters will it clear the fence? If not, where will the baseball strike?

2. Homework Equations

dv=1/2*at2
dh=Vh*Δt
Kinetic equation d=Vi*t+ 1/2*at2
dh=-V2*sin2θ /g
sinθ=opp/hyp
cosθ=adj/hyp
Δt=-2Vsinθ / g

3. The Attempt at a Solution

dh= cos(45)*35=24.7m
Δt=24.7m/35m/s=0.71s

d=(35)(0.71)+1/2(9.8)(0.71)2=27.3m

No, the baseball will not clear the fence.

I am really lost on this problem just because we are given so many measurements to work with and I don't know the appropriate time to use them and which equation is the right one. If someone could please check my work and correct me and also verify my significant figures that would be greatly appreciated!Thanks so much in advance.

 

[jimbobian]

Ok, firstly I don't get the answer you get and secondly, you're right there are a lot of equations you've got written down there but you need to home in on this one, which is the one that will solve it:

d=Vit + 1/2(at^2) [which I'm going to rewrite replacing Vi with u (personal preference ;) )]

So d=ut+0.5at^2

You can apply this equation vertically and horizontally and by doing each in turn (but I'm leaving you to get the order) you should be able to get the answer.

 

[dani123]

did i use the right equation to find t in the beginning?

 

[dani123]

I re-did the problem and used a different approach after consulting my textbook and got that the ball would clear the fence by 1.4m. Does this seem right?

The known things in this problem are:
|V_i|=35.0m/s
θ=45°
Δx=120m
h=4m
a=-9.8m/s²

Unknown:
Δt, V_iy, V_ix, Δy

I started by finding the horizontal component: V_ix=|V_i|*cos(45)=24.7m/s

vertical component: V_iy=|V_i|*sin(45)=24.7m/s

find t interval: Δt=Δx/V_ix=120m/24.7m/s= 4.86 seconds

find h of ball at time it reaches fence:
Δy=V_iy*Δt+1/2a*Δt²= 4.387m

y_fence=h-y_{ground to x-axis}=4m-1m=3.0m

∴cleared fence by 4.387m-3m= 1.4m

 

[jimbobian]

That's not what I got as the final answer, but some of the intermediate working is definitely correct.
EDIT: As I worked through your post, I discovered the only mistake was rounding errors. Read on to find out more!

V_ix = 24.7m/s - Correct
V_iy = 24.7m/s - Correct
time to travel 120m in x-direction = 4.86s - I get 4.85 but you've only made a rounding error (by using 24.7 rather than 35Cos(45)) so in essence correct although you will start to lose accuracy rapidly
Equation to find change in y - Correct

But now the rounding errors that have been propagating through your work have made a substantial difference to the answer. If you hold non-rounded numbers right the way through to this point the value for change in y should be exactly 4.8m

As for the final part of your reasoning, the technique is absolutely correct. You have remembered to correct for the ball starting 1m above the plate (which many people forget to do) and have calculated the height of the top of the fence in your y-axis (and have done so correctly). Once again, what's let you down is the rounding errors carried over from the ball's height.

To get to the final answer that I have, you need to follow the logic you used again either:

1) Not calculating any values (sin, cos, exponents) and plugging equations into each other and then calculating an answer at the very end

OR

2) My preferred technique - Make use of the memory functions on your calculator. Scientific calculators will usually have at least 7 different variables you can use to store intermediate answers - Mine has A B C D X Y Z. When you calculate the x-velocity of flight, store it in A. When you then find the time, use A in your calculation and store time of flight in B. Etc etc. to the final answer

OR

3) Write down the intermediate calculations with more significant figures than your final answer will need. Write down 6, 7 or maybe all 10 of the digits your calculator gives you and then use this in your next calculation.

Cheers.

 

[dani123]

Thank you so much for your help and time! so the final answer should be that the ball cleared the fence by 1.8m?

 

[jimbobian]

Yes! No problem