(II) Police investigators, examining the scene of an accident ...

Question

(II) Police investigators, examining the scene of an accident involving a car and an old truck, measure 72-m-long skid marks for the truck, which nearly came to a stop before colliding with the car at rest. The coefficient of kinetic friction between rubber and the pavement is about 0.80. Estimate the initial speed of the truck assuming a level road.

Accepted Answer

Hey, everyone in this problem, a man is riding a bicycle in a field, seeing an obstacle, he hits the brakes as a result, the wheels get locked and the bicycle is going to slide a distance of 15 m on the ground before it stops. We're told that the coefficient of kinetic friction between the material of the bicycle tires and the ground is 0.5. And we're asked to calculate the initial speed of the bicycle. We're told to assume the ground is perfectly level. We have four answer choices all in meters per second. Option A 1.2 option B 12, option C 24 and option D 150. So let's start. We're gonna start with a free body diagram. We're giving some information about the coefficient of kinetic friction. So we want to think about all of the forces acting on this bicycle. And then we're gonna write out all of the other things we're given in this problem. So we have our bike, I'm just gonna draw it as a point, it's on the ground. So we know that there's gonna be a normal force acting straight upwards. We know that the force of gravity acts downwards. And we're gonna assume this bike is moving to the right, the bike is moving to the right. That means that the force of kinetic friction is gonna act to the left. Ok. The friction is going to oppose the motion. So it's acting in the opposite direction. We're gonna call that FK because it's kinetic friction. These bike tires have been locked, so they're just sliding along the ground. So we're dealing with kinetic friction there. Now we're gonna take up into the right as our positive direction in our free body diagram and for our calculations and now we can think about these other things we've been given as well. And what we've been given is, well, we wanna find the initial speed we'll call that V not, that's what we're looking for. We know that the final speed is zero. This bike is going to come to rest. So we have 0 m per second. You know, it's gonna travel 15 m before it comes to a stop. So delta X is equal to 15 m, but we don't have any information about the acceleration or the time that this takes. So if we want to think about our kinematic or U AM equations, we only have two known values and that's not enough to solve for this unknown of V. Not. So what we're gonna do is use our free body diagram, use the information we have about the forces to find our acceleration. We know we can relate the acceleration to the forces through Newton's Second law. Once we have this acceleration, we'll have three known values and we can use our kinematic U AM equations to find Von. And so that's what we're gonna do. And we're gonna start by looking at the sum of the forces in the X direction and we're gonna look in the X direction because that is where our acceleration acts. OK. So Newton's second law tells us that the sum of the forces in that X direction, that direction of motion is gonna be equal to the mass multiplied by the acceleration in the X direction. The only force we have acting is a force of kinetic friction and that's acting in a negative direction. So we get negative FK is equal to ma hey, we want to solve for the acceleration. So we can divide both sides by the mass. We get that the acceleration will be negative the force of kinetic friction. They recall that the force of kinetic friction is given by the coefficient of kinetic friction in the UK multiplied by the normal force. And OK. And this is all divided by that mass M. So we're getting there and we have this expression for acceleration. The problem is we don't know this normal force. And so in order to calculate the acceleration and use it in our kinematic equations, we have to find this normal force N first, now that normal force and acts in the Y direction. So we need to switch gears and look at the sum of forces in the Y direction and that some of the forces in the Y direction in this case will just be equal to zero. If this bike is not moving up and down, off the ground, it's not moving in the vertical direction. So we get that sum being equal to zero. Now, in the positive Y direction, we have the normal force acting in the negative Y direction, we have the force of gravity. So N minus FG will be equal to zero. It tells us that our normal force will be equal to the force of gravity. And that force of gravity recall is given by the mass multiplied by the acceleration due to gravity. G so yeah, N is equal to MG. So we can substitute this expression back into our expression for the acceleration. The acceleration will be equal to negative mu K multiplied by MG divided by M. OK. So that MG coming from that normal force we calculated we can see that the mass M will divide out from both of these terms or both the numerator and denominator I should say. And so we get our acceleration is just going to be negative UK multiplied by G. We can substitute in values. We get that A will be equal to negative 0.5 multiplied by 9.8 m per second squared, which gives an acceleration of negative 4.9 m per second squared. And this makes sense that this is negative because we know that our bike is slowing down, the acceleration is negative. We are decelerating. This makes sense. We're gonna put a red box around that and that's gonna help us. So let's go back to the top where we wrote out everything we were given in the problem. We needed to have an extra value to solve our U AM equations. And now we do, we know that A is negative 4.9 m per second squared. So what we can do is choose our kinematic or U AM equations with the three known values we have VF delta X and A as well as the unknown. We're looking for V not. And that equation is going to be VF squared is equal to V not squared plus two A delta X. We can substitute in our values zero meters squared per second squared will be equal to V knot squared plus two, multiplied by negative 4.9 m per second squared, multiplied by 15 m. If we work out everything on the right, the second term on the right hand side and we move it over, we get that VNO squared will be equal to 36.75 meters squared per second squared. And taking the square root on both sides gives us that V knot is approximately equal to 12.1244 m per second. Ok, so that is the initial speed that this bike was moving before it started to come to rest before it hit the brakes. If we compare this to the answer choices we were given when we round to two significant digits, we can see that the answer corresponds with option B 12 m per second. Thanks everyone for watching. I hope this video helps see you in the next one.

Struggling with Physics?

Watch next

Kinetic Friction practice set