Force and motion practice questions Put another way: 1. A cart with very low friction wheels (pretend that there is no friction) has a mass of 750 grams (this is the mass of the cart and the fan combined). The fan on the cart can bring the cart from rest up to a speed of 1.2 m/s in 3.0 seconds. a. Under the conditions just described, what is the acceleration of the cart? The change in velocity was 1.2 m/s and this happened in a time period of 3.0 seconds. b. How long would it take the same fan to bring the same cart up to a speed of 1.8 m/s? 1.8 m/s is faster than 1.2 m/s by a factor of 1.5. So assuming the same fan would produce the same acceleration, it would take 1.5 times as long to get up to that speed. c. If the cart starts out with a speed of 0.6 m/s, how long will it take to get it up to a speed of 3.0 m/s? What matters is the change in velocity, not the velocity. The change in velocity is 3.0 m/s - 0.6 m/s = 2.4 m/s. That's exactly twice as fast as in the original experiment (taking 3.0 seconds), so it would take twice as long, or 6.0 seconds. d. If we could somehow get the fan to push the cart twice as hard, how long would it take to get this cart from rest up to 1.2 m/s? This is the same speed as in the original equation, but if the fan pushes twice as hard we should have twice as much acceleration which is 2 x 0.40 m/s 2 = 0.80 m/s 2. (exactly half as long) e. If a 250 gram weight is added to the cart, how long would it take the original fan (not the one that pushes twice as hard) to get this more massive cart from rest up to a speed of 1.2 m/s? Now the mass has been increased so the acceleration must decrease. This means it will take longer to get up to that speed.
Position in meters Velocity in m/s 2. The graphs below show the motion of a cart as a function of time during an eight second period: 14 12 10 8 6 4 2 0 Position as a function of time 0 1 2 3 4 5 6 7 8 Time in seconds 3 2.5 2 1.5 1 0.5 0 Velocity as a function of time 0 1 2 3 4 5 6 7 8 Time in seconds a. Using only the graph on the left, how would you find the average velocity of the cart during those eight seconds? (looking for an explanation here) Find the total change in position and divide it by the total time taken. This is the same as finding the slope of a straight line that connects the ends of that curved graph. b. Using only the chart on the left, what was the average velocity during those eight seconds? (looking for a number with units here) c. Does your answer from part b look reasonable in light of the graph on the right? Explain. Yep. The average velocity would have to be somewhere between the extremes of velocity in the velocity graph and since that graph is a straight line, it makes sense that the average would be the velocity right in the middle, at 4.0 seconds. That velocity is 1.5 m/s. d. Using the graph on the right, what was the acceleration of the cart during those eight seconds? The acceleration is the slope of the velocity vs. time graph. Since the graph is a straight line, you can find the slope from any two points on the graph. I'll use the velocities at 8 s and 4 s. e. If the cart had a mass of 500 grams, what was the unbalanced force on the cart in newtons? (careful, that says 500 grams) Wow. Does that say grams? That really almost tripped me up. The first thing you have to do is convert to consistent units and the only consistent units we know are newtons, meters, and kilograms. So we need to know that 500 grams is 0.5 kg.
3. Katrina and Alexei are ice skaters. Alexei has a mass of 72 kg. Katrina has a mass of 54 kg. At one point in their pairs skating routine, Alexei skates up behind Katrina, they stand dramatically still for a moment and then he pushes firmly on her back. (Assume that they are both good enough skaters that neither one of them falls down as a result!) a. Alexei pushes on Katrina's back for 0.90 seconds with a constant force of 96 newtons. Assuming we can pretend that there is no friction between the skates and the ice, how fast would Katrina be moving at the end of those 0.90 seconds? The unbalanced force is 96 newtons. Her mass is 54 kg. So Her change in velocity will be this acceleration times the time: b. Under the same assumptions, what is happening to Alexei during those same 0.90 seconds? Why? He would accelerate in the other direction because she pushes on him just as hard as he pushes on her. c. Under the same assumptions, what would Alexei be doing at the end of those 0.90 seconds? (You should come up with a number if you can.) His acceleration is less since his mass is greater. As a result his final speed will be less by the ratio of their masses (and he will be moving in the opposite direction). d. It isn't really reasonable to assume that there is no friction between the skates and the ice. There is less friction than there would be, say, between sneakers and a tile floor, but not zero friction. It turns out that during the time Alexei pushed her, there was a friction force of a little less than 30 newtons on her skates in the direction opposite the push. What was the effect of that friction force? You don't need to calculate how fast she was going. Simply explain in words what is the result of a 96 newton force in one direction when there is a 30 newton force in the other direction. Since the friction force points in the direction opposite the way that he was pushing, it has the effect of diminishing the unbalanced force by 30 newtons. The actual unbalanced force would be 66 N, not 96 N.
e. After the 0.90 second push, Katrina was actually moving with a speed of 1.1 m/s. (Give yourself a gold star if you can prove it.) She slides away from Alexei at a speed of 1.1 m/s, but she has a force of 30 newtons pointing in the opposite direction. Assuming that this friction force is the only force that matters at this time, what is the effect of the friction force on her motion? Her acceleration would be reduced by the ratio of 66 N (the actual unbalanced force) to 96 N (what we originally assumed to be the unbalanced force). Since the amount of time doesn't change, her velocity would be reduced by the same ratio. I want my gold star: (so there!) f. Under the same assumptions as in part e, after Alexei quits pushing her and she begins to glide away with a speed of 1.1 m/s with a friction force of 30 newtons in the other direction, how long does it take before she stops? Who wrote these problems, anyway? Now Alexei isn't pushing so the unbalanced force on her really is 30 newtons. Her acceleration will be She starts out moving 1.1 m/s and she slows down by the amount 0.55 m/s every second after that. So the amount of time it takes her to stop is... (do you understand the units?) g. If it was Alexei sliding away at a speed of 1.1 m/s with a force of 30 newtons in the other direction, would it take more time or less time for him to come to a stop? Why? More time! He has the same speed and the same force acting on him but he has more mass so he has more (wait for it) inertia! Since he has more inertia it will take longer for him to stop under dentical conditions.
4. In the musical, "Man of La Mancha", Don Quixote's friend Sancho Panza was describing what happens when a stone collides with a pitcher (meaning a ceramic or glass container for water, not a baseball pitcher). He said, "Whether the stone hits the pitcher, or the pitcher hits the stone, it's going to be bad for the pitcher." Newton might have agreed, but... a. Is this true because the stone hits the pitcher harder than the pitcher hits the stone? (Explain in terms of Newton's laws.) Nope. Newton's third law says the force that the stone exerts on the pitcher has got to be the same size as the force that the pitcher exerts on the stone. Yes, we should call it Newton's third "theory" but even so we would need to remember that scientists and engineers put the third law to the test millions of times every year and no violation has ever been observed. In this case things turn out worse for the pitcher because the pitcher is more fragile than the stone. (It might or might not have less mass, depending on the pitcher and the stone used for the experiment.) b. Would Newton have agreed that it doesn't matter whether the stone hits the pitcher, or the pitcher hits the stone? Yes. It makes no difference how you describe the collision. Newton might have gone so far as to say that these words are misleading because both stone and pitcher must hit each other simultaneously. 2. In the game of bowling, a bowling ball can knock down ten pins which go flying in all directions while the ball keeps moving through. a. Why is there this difference between the behavior of the ball and the behavior of the pins? The pins have less mass, typically less than about 1/4 the mass of the bowling ball. (Also, the ball is moving pretty fast so a change in the velocity of the ball is much less dramatic than a change of the same size in the velocity of a previously stationary pin.) b. Do you think the bowling ball changes velocity at all when it strikes a pin? Yes. Newton's third law says that both objects exert forces on each other and the second law says that these forces will result in changes in velocity of each object. c. When a 7.0 kg bowling ball hits a 1.5 kg bowling pin with a force of 350 newtons, the 1.5 kg bowling pin hits the bowling ball with a force of... what? (How great is the force that the pin exerts on the ball?) 350 newtons. It has to be the same size as the force that the ball exerts on the pin. The third law hasn't failed once in 320 years. So think about this. Newton's laws really are theories. The theories laid the foundation for the industrial revolution. You trust your life to the accuracy of these theories every time you get into an elevator, a car, or a plane. They explain tides, heart rates, and the weather. Think about that the next time you hear somebody say, "It's just a theory."