(II) You buy a plastic dart gun, and being a clever physics st...

Question

(II) You buy a plastic dart gun, and being a clever physics student you decide to do a quick calculation to find its maximum horizontal range. You shoot the gun straight up, and it takes 3.4 s for the dart to land back at the barrel. What is the maximum horizontal range of your gun?

Accepted Answer

Hey everyone in this problem, a curious inventor constructs a toy rocket launcher and decides to calculate the maximum horizontal range of the toy rocket that it launches after setting it up, they launch a toy rocket vertically up and it takes 4.5 seconds for the rocket to glide back down to the launcher. We're asked to find the maximum horizontal range of the toy rocket. We have four answer choices in this question. All in meters. Option A 225 option B 216 option C 49.6 and option D 22.5. All right. So we have kind of two parts to this question. The first part is that we have our rocket launcher. So pretend that this is a rocket and it's gonna be launched straight upwards, vertically upwards and it's gonna come back down to the position. It was launched in, in 4.5. Ok. So this case, this is gonna give us some information about the speed at which the rocket launcher can launch this rocket. Then we're gonna tilt that rocket launcher at some angle and we don't know what this angle is, and we wanna figure out this maximum range that it can achieve. And we're gonna call that our max that maxim horizontal. I recall that our range are, it's equal to V not squared multiplied by sign up to the all divided by the acceleration due to gravity G. So if we're looking at this equation and we wanna find the maximum, OK, the maximum of this equation, the maximum R value we can have is going to occur when sy of two theta is equal to one. And we know that sign takes a maximum value of one V not squared. OK. That's gonna be an initial value that's already determined and it's squared. So that's a positive value. And then G that's a set constant acceleration due to graph. OK. So the maximum of this entire value R is gonna be one sign if two theta is equal to one, this tells us that two theta will be equal to 90 degrees. If we restrict ourselves to a 90 degree angle, we're launching this rocket somewhere forward. And so theta is gonna be equal to 45 degrees. No, we weren't really asked about this angle, but that gives us some more information. We're gonna get this maximum horizontal range when we launch this at a 45 degree angle, which makes sense. OK. That right in the middle. All right. So if we go back to our equation for the range, then we know that the maximum range will just be equal to V not squared divided by G. Again, we know the acceleration due to gravity but we not, we don't know, we don't know the speed at which the rocket is being launched, but we can figure it out using this first case when we launch that rocket straight up. So let's look at that case, when we launch that rocket straight up, it's gonna have some initial velocity which we'll call V. Not Y we wanna find that we don't know the final velocity. We do know that we have acceleration due to gravity. We're gonna take up to be positive, our acceleration due to gravity acts downwards. And so it's gonna be negative 9.8 m per second squared. We know that delta Y is equal to 0 m. OK? That displacement is zero. We're starting and ending in the same place. So the final position minus the initial position is just zero. And we know that this takes 4.5 seconds. So we have three known values we wanna calculate V not, we're gonna use the U AM equation with the four values of interest. The three that we know the one we're, and that equation is gonna be delta Y is equal to V, not Y multiplied by T plus one half A T squared. Substituting in our values, 0 m is equal to V not Y multiplied by 4.5 seconds who lost one half multiplied by negative 9.8 m per second squared, multiplied by 4.5 seconds squared. We can move this entire second term to the left hand side. OK. Just the number we can simplify it and then we're just gonna kind of reverse the order and we can write this as V not Y multiplied by 4.5 seconds is equal to 99.225 m. Final step dividing by 4.5 seconds. V not Y is going to be equal to 22.05 m per second. All right. So we know the speed that this rocket launcher launches rockets at. Ok. Now, in this case, it's that initial Y velocity, but that's the total velocity or the total initial speed, we're launching it straight upwards. So that is the launch speed. When we tilt it, it's still launching at that same speed and the speed is now on an angle, but it still has the same initial speed. So this value is just gonna be the V knot value for our R max equation. So we can get back to that R max equation. That's what we're really looking for that maximum range. And we have that R max will be equal to 22.05 m per second, all squared divided by 9.8 m per second squared, which gives us a maximum range of about 49.6125 m. We can compare this to the answer choices we were given. We're gonna round to three significant digits and we can see that the correct answer is going to be option C 49.6 m. Thanks everyone for watching. I hope this video helped see you in the next one.

Struggling with Physics?

(II) You buy a plastic dart gun, and being a clever physics student you decide to do a quick calculation to find its maximum horizontal range. You shoot the gun straight up, and it takes 3.4 s for the dart to land back at the barrel. What is the maximum horizontal range of your gun?

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