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Ball Thrown Up An Inclined Plane

A yellow ball is thrown with initial speed \(v_{0} = 2\) \(m/s\) up an inclined plane. The plane is inclined at an angle \(15 ^ \circ\) above the horizontal, and the ball's thrown with an angle \(45 ^ \circ\) above the plane. If \(R\) denotes the ball land's distance, find the value of \(R\). (assume \(g = 10\) \(m/s^2\))

I found that the answer is \(\frac{2}{5}(\sqrt{3} - 1)\) \(m\), but I'm not sure. Could anyone clarify? Thanks :)

Note by Nabila Nida Rafida
4 years, 1 month ago

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\[v_{x} = 2 \cos 60^\circ = 1\]

\[v_{y} = 2 \sin 60^\circ = \sqrt{3}\]

\[x = v_{x}t = t\]

\[y = v_{y}t - \frac{1}{2}gt^2 = \sqrt{3}t - 5t^2\]

Because \(\tan 15^\circ = \frac{y}{x},\) then \(x = t = y(2 + \sqrt{3})\)

By substituting \(t = y(2 + \sqrt{3})\) to \(y = \sqrt{3}t - 5t^2,\)

\[y = \sqrt{3}(2 + \sqrt{3})y - 5(2 + \sqrt{3})^2y^2\]

\[y(5(7 + 4\sqrt{3})y - 2(1 + \sqrt{3})) = 0\]

Since \(y \neq 0,\)

\[y = \frac{2(1+\sqrt{3})}{5(7+4\sqrt{3})}\]

Because \(\sin 15^\circ = \frac{y}{R},\) then \(R = \frac{y}{\sin 15^\circ}\)

Hence, \[R = \frac{4\sqrt{2}(2-\sqrt{3})}{5}\]

Fariz Azmi Pratama - 4 years, 1 month ago

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Firstly, good question Nabila and great effort on your answer.

Fariz nice approach to the question, I completely agree with your answer as I verified it using a different method; which was finding the intercepts of the trajectory of the projectile and the plane.

I found using simple physics formulae that the height is \(0.15m\)

I then used this to model a formula for the trajectory; \(y=ax^{2}+bx\) where b is the initial gradient of the projectile \(tan(\pi/3)\),

Using height of \(0.15\), \(a\) was found to be \(-5\)

Hence, \(y=\sqrt{3}x-5x^{2}\) is the trajectory in the Cartesian coordinate system

This is now intercepted with the plane which is linear; \(y=bx\), were b is \(tan(\pi/12)\)

Therefore the plane is represented as; \(y=(2-\sqrt{3})x\)

Furthermore, the coordinates of the intercepts of the trajectory and the plane are calculated to be:

\(y=\frac{6}{5}(\sqrt{3}-2)\) and \(x=\frac{2}{5}(\sqrt{3}-1)\)

Finally, using Pythagoras's Theorem, \(R=\sqrt{(\frac{2}{5}(\sqrt{3}-1))^{2}+(\frac{6}{5}(\sqrt{3}-2))^{2} }\)

Hence, \(R=\frac{4}{5}\sqrt{2}(2-\sqrt{3})\)

Joel Jablonski - 4 years, 1 month ago

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Nice one!

Nabila Nida Rafida - 4 years, 1 month ago

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Mathematical approach, cool :)

Fariz Azmi Pratama - 4 years, 1 month ago

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Thanks, it seems my answer yet completed

Nabila Nida Rafida - 4 years, 1 month ago

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Nice solution, though just a slight mistake in the end. I believe, you can easily get the right answer if you try to deduce a point in the trajectory where \( \frac {y} {x} = \tan 15^\circ \) using the above equations. This would give the required R most probably.

Lokesh Sharma - 4 years, 1 month ago

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Would you like to fix it, Lokesh? I'm not sure about my answer.....

Fariz Azmi Pratama - 4 years, 1 month ago

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@Fariz Azmi Pratama Referring to your solution, equation (3) and equation (4) define the trajectory of ball. The ball lands at the point \( (x, y) \) such that \( \frac {y} {x} = tan 15^ \circ \).

Solving these three equations, the point can be located and hence the distance where the ball lands. Diving equation (4) with equation (3), \( \frac {y} {x} = \sqrt3 - 5t \) and using \( \frac {y} {x} = tan 15^ \circ \), finally \( \sqrt3 - 5t = tan 15^ \circ \Rightarrow \sqrt3 - 5t = 2 - \sqrt(3) \).

Solving this we have \(x = t =\frac {2 \sqrt3 - 2}{5} \). Using trigonometry we get \(\frac{x} {R} = cos 15^ \circ\) and hence \(R = \frac{ x }{ cos 15^ \circ} \Rightarrow R = \frac {4(2 \sqrt3 - 2)}{5(\sqrt6 + \sqrt2)} \).

Finally, \( R = \frac {4(2\sqrt2 - \sqrt 6)}{5} \) which is same as yours. Hahaha. I was wrong about your solution. Sorry about that one.

Lokesh Sharma - 4 years, 1 month ago

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@Lokesh Sharma Haha it's okay.

Fariz Azmi Pratama - 4 years, 1 month ago

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\[Range=\frac{ 2*\sin45 }{ 5 *\cos^215 }\]

Cody Martin - 4 years, 1 month ago

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We need to start by finding final speed, \(v_f\).

After that, u can start using conservation of mechanical energy

\(K_i + U_i = K_f + U_f\)

\(\frac{1}{2}mv_0{}^{2} = \frac{1}{2} mv_f{}^{2} + mgR\sin 45^{\circ}\)

\(v_0{}^{2} = v_f{}^{2} + 2gR\sin 45^{\circ}\)

Now u have all the values except \(R\) and u can easily find \(R\)

Saad Haider - 4 years ago

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What is the value of \(v_f\)?

Lokesh Sharma - 4 years ago

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U can use the equation of kinematics

In the y-direction

\(v_{yf}^{2} = (v_0\sin45^{\circ})^{2} -2gR\sin45^{\circ}\)

\(v_{yf} = v_0\sin 45^{\circ} - gt\)

\(R\sin 45^{\circ} = v_0\sin 45^{\circ}t - \frac{1}{2} gt^{2}\)

\(R\sin 45^{\circ} = \frac{1}{2}(v_{yf} + v_0\sin 45^{\circ})t\)

In the x-direction

\(R\cos 45^{\circ} = v_0\cos 45^{\circ}t\)

Using the equations, we can find \(v_{yf}\). We also have \(v_{xf} = v_{xi} = v_0\cos 45^{\circ}\). Then u can find \(v_f\) by Pythagoras theorem: \(v_f = \sqrt{v_{xf}^{2}+v_{yf}^{2}}\)

Saad Haider - 4 years ago

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rotate Oxy 15

gx= -10sin15; gy= -10cos15

t = 2vsin45/gy

R=vcos45t - 1/2 * gx * t^2 =2*(1 - tan15)/cos15

Thanh Nguyen - 4 years ago

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tan15=y/x =(2.sin60.t-1/2.10. t^2)/(2.cos60.t)
2-sqrt{3} =sqrt{3}-5t
t={2/5}(sqrt{3}-1)
R= x/sin 15 =t/sin 15 = {4/5}sqrt{2}

Viji Raj - 4 years, 1 month ago

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R = x / cos 15

Lokesh Sharma - 4 years, 1 month ago

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Yes you are right Lokesh. R=x/cos15 that gives me the same answer as yours

Viji Raj - 4 years, 1 month ago

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\[\frac{ 4 }{ 5 }(2\sqrt{6}-3\sqrt{2})\] ??

Cody Martin - 4 years, 1 month ago

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How did you get the answer?

Fariz Azmi Pratama - 4 years, 1 month ago

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yes.I agree

David L. - 4 years, 1 month ago

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Hey, I think what you found was incorrect in the sense that what you found wasn't what the question was asking for. I found that your answer was what I got for the horizontal distance, not the actually length between the ball land's distance. What I did was find the equation for both lines (the inclined plane and the trajectory) find the intersection which is where you got, and the last bit is apply the Pythagorean theorem to get that length.

Victor Phan - 4 years, 1 month ago

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it should be 4 (sqrt2)/5

Viji Raj - 4 years, 1 month ago

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Yes Victor is right. You may even use trigonometry sin 15 = (sqrt3-1)/2

Viji Raj - 4 years, 1 month ago

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i think it would be R=uusin2*/g=0.4m

Abhimanyu Manu - 4 years, 1 month ago

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