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Two cannons are mounted as shown in the drawing and rigged to fire simultaneously. They are used in a circus act in which two clowns serve as human cannonballs. The clowns are fired toward each other and collide at a height of $1.00 \mathrm{~m}$ above the muzzles of the cannons. Clown $\mathrm{A}$ is launched at a $75.0^{\circ}$ angle, with a speed of $9.00 \mathrm{~m} / \mathrm{s}$. The horizontal separation between the clowns as they leave the cannons is $6.00 \mathrm{~m}$. Find the launch speed $v_{0 B}$ and the launch angle $\theta_{\mathrm{B}}\left(>45.0^{\circ}\right)$ for clown $\mathrm{B}$.

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$\theta_{B}=81.45^{\circ}$$v_{0, B}=8.791 \mathrm{m} \cdot \mathrm{s}^{-1}$

Physics 101 Mechanics

Chapter 3

Kinematics in Two Dimensions

Motion in 2d or 3d

Rutgers, The State University of New Jersey

Simon Fraser University

McMaster University

Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

10:12

A vector is a mathematical entity that has a magnitude (or length) and direction. The vector is represented by a line segment with a definite beginning, direction, and magnitude. Vectors are added by adding their respective components, and multiplied by a scalar (or a number) to scale the vector.

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Two cannons are mounted as…

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A $1400 \mathrm{~kg}$ cann…

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A 20.0 -kg cannonball is f…

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A 20.0 -kg cannon ball is …

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07:24

Two cannonballs are shot f…

06:05

A cannon, located 60.0 $\m…

05:21

04:21

A 70.0 -kg circus performe…

02:37

Two cannons are fired from…

04:16

06:30

So the question states that two clowns were being shot out of a cannon, one of one's being shut out at nine meters per second at a 75 degree angle. And the other one, we don't know his speed or the angle he shot out at. But we are told that the cannons are both six meters apart and the clowns hit each other at one meter in the air. So the first thing we really need to do is figure out how long it takes for the clown going nine meters per second to reach one meter, uh, one meter in the air. So to do this, we can use our king a magic formula, which is that the change in the UAE displacement is equal to the initial velocity in the Y direction times the time plus 1/2 times the acceleration times time script. So we can figure out what the initial velocity is in the Y direction by using trigonometry. So we know that the sign of 75 degrees should be equal toothy vertical component of the velocity over nine. So that means Thea vertical component of the velocity must be equal to nine times the sign of 75 degrees. So we know what V not is. We know what Delta Y is, and we know what a is. It's the acceleration due to gravity. You said we conflict all this in so one meter is equal to nine times sign of 75 degrees times t plus 1/2 times the acceleration, which is negative 9.8 times t squared so we can simplify this equation a little bit. So we'll have negative 9.4 point nine t squared, plus nine times sign of 75 degrees times T minus one should be equal to zero. So we can either use quadratic formula to solve this, or we can use a graphing calculator, which is what I did. And when we plug this in, we find that the roots are either equal 2.124 or 1.651 seconds. Uh, the route we should use in this case because they're telling us that the angle needs to be greater than 45 degrees should be the one that is Ah, at a later time. If we choose the earlier time, the angle is likely going to be below 45 degrees. So we need to choose 1.65 so t equals 1.651 Now we know the time that it takes for the clown to get toe one meter. We can figure out how far to the right he is shot in this time t So his horizontal velocity visa vax he's going to be Ah, we can find that because using trigonometry because co sign of 75 degrees is the, um, visa Becks over nine meters per second. So Visa Becks has to equal nine times co signer 75 degrees. And so let me write that down respects is nine times co sign 75 degrees and we know that the equation that describes the range of the projectile motion is the horizontal velocity times the time should be equal to the range, so we know the horizontal velocity and the time, so that's pointless. In nine times co sign 75 degrees times 1.651 seconds. That means that the total displacement of the clown shot out at nine meters per second is three point 845 eight meters to the right So now that we know this, that means that the clown that is shot on the right side, where we don't know the velocity or the angle, it means that his his displacement must must be six minus this number. So the clowns displacement being shot from this other side must be equal to six minus 3.8458 which is equal to 2.154 two meters. So now that we know the, um the distance he needs to travel and we also know the time he needs to travel t get to make sure that they both hit at one meter in the air at the same distance. Um, we can use our equation that states that the velocity in the X direction times a time must be equal to the displacement. We know what the time is. It should be 1.651 seconds. We're trying to find the his horizontal velocity, and we know the distances. Travel is 2.15 for two meters. So when we saw for vfx, we get the V of X is equal to 1.30 for seven. No, the next thing we need to do is figure out what his vertical velocity is equal to. So to do this, we can use another Kinnah Matic equation which states that his change in his ah vertical displacement is equal to his initial velocity in the Y direction times the time plus 1/2 times the acceleration times a time squared. So we know his displacement has to be equal to one meter. We're trying to find his vertical component of of lost e. We know the time it takes is 1.651 seconds and we know the acceleration is to be negative. 9.8, Do you do gravity And his time once again is 1.651 seconds and this is going to be squared. So when we subtract 1/2 times negative 9.8 times 1.651 squared on both sides and then divide by 1.65 one on both sides we get that his vertical component for the velocity ah is equal to eight point six 955 meters per second. So now that we know his, uh, horizontal velocity as well as his vertical velocity we confined his total his total the magnitude of the velocity because we know that total magnitude is equal to the square root of the horizontal component, plus the vertical component. And when we plug this in, when you get that, his total magnitude is equal to 81 point for seven meters per second. Actually, I'm sorry. That's my bed. It should be. Instead of 81.47 it should be eight point 79 meters per second, equals 8.79 meters per second. And if we want to find the angle that the clown has to make, we just need to take the 10 inverse of the vertical component divided by the horizontal component of the velocity. And when we do this, we get a state of value of eight. 81 point for seven degrees, and that's the final answer

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