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The result we got from the experiment is that as the total mass of the tub increases, the shorter the distance that it slides along the floor. Why is this so? Let’s assume that there are two tubs with different weights in them and find out the reason why the heavier tub travels a shorter distance than the lighter tub.

There are three stages in moving the tub with a rubber band. The first stage is when the rubber band hasn’t been released and is pulled back, the second stage when the rubber band is released and is going back to its original straight position, the third stage is when the rubber band is straight and the tub is going forward on its own.

We can use two methods to explain it, one with force and the other with energy. How does forces explain that the heavier tub travels a shorter distance?

Firstly, in the first stage, the rubber band has not been released, so there are no forces acting on the tubs. Then, the rubber band is released and is going back to its original straight position. Let’s assume that the force applied onto the tubs with the rubber band is F. As the force is applied with the same rubber band and the distance that it is pulled back is fixed, the force it applied will be the same no matter what the mass the tub in front of it is. Besides the force applied by the rubber band, there is also frictional force. The frictional force is calculated the equation F = μmg, which means that the mass of the tub is directly proportional to the friction between the tub and the floor. So the heavier tub is going to have more friction than the less heavier tub. Assume the friction of the heavier tub and the lighter tub be f1 and f2 respectively. The resultant force R1 applied on the heavier tub is (F-f1) N and the resultant force R2 applied on the lighter tub is (F-f2) N. As f1 > f2, R1 < R2. According to Newton’s Second Law of Motion, F=ma. The lighter tub has a bigger resultant force, so it is going to have a larger acceleration.

A larger acceleration means that the lighter tub is going to reach a higher velocity when it leaves the rubber band and begins to travel on its own, entering the second stage.in the second stage, there is only one force acting: the frictional force. As mentioned before, the formula of friction is F = μmg and Newton’s Second Law of Motion is F=ma. After substituting the second formula into the first, we get ma = μmg, and after rearranging it, we get a = μg. This means that the acceleration of the heavier and the lighter tub in the second stage does not relate to their mass, so they have the same acceleration slowing down. While their acceleration is the same, the velocity of the lighter tub is higher. The equation of acceleration is: a=(V_2- V_1)/∆t,by rearranging it we can get: t=V_(2-V_1 )/a. In this case, the value of V2 is 0, and the acceleration is a negative value, so t ∝ V, where t is the time the tubs need to reduce its speed to 0 and V is its current speed. As a result, the lighter tub will need to travel a longer time before it stops. The formula that connects the acceleration, time and distance together is d = 1/2at^2. In this case, the value of the heavier tub and the lighter tub is the same, so the longer the time it takes, the further the distance. Since the lighter tub needs a longer time to slow down, it will travel a longer distance than the heavier tub.

Another way to explain this is with energy.

In the first stage, the only energy is the elastic potential energy in the rubber bands. Elastic potential energy (measured in the unit joules) is equal to ½ multiplied by the stretch length ("x") squared, multiplied by the spring constant "k." As the stretch length of the rubber band in the experiment is a fixed value and the spring constant is also fixed as we are using the same rubber band, the elastic potential energy before the release of the rubber band is always the same no matter what the mass of the tub in front of it is. As a result, we can have this equation: P1 = P2, where P1 and P2 are the elastic potential energy of the rubber bands with tubs of different masses in front of them.

In the second stage, the rubber band is released and is going back to its original straight position. In this stage, the potential energy of the rubber bands is transferring into two energy: the kinetic energy of the tub and the work done by friction between the tub and the floor. How does the mass of the tub affect the work done by friction? The equation to calculate the work done by friction is f × d, the frictional force times the distance the object travels. The distance the tub, heavy or light, travels when the rubber band went from a pulled-back position to a straight line is a fixed value. The heavier of the two tubs is going to have more friction than the less heavy tub as it has more mass. As a result, the work done by the friction of the heavier tub is larger than of the lighter tub. To find the relationship between the kinetic energy of the two tubs, we can have the following equations: ∵ P1 = P2, ∴ K1 + W1 = K2 + W2, where K1 and W1 are the kinetic energy and the work done by the friction of the lighter tub and K2 and W2 are of the heavier tub. As W1 < W2, K1 > K2.

In the third stage, after the rubber band is straight and the tub is going forward on its own, there is no more potential energy transferring into kinetic energy, so the only forces left are the kinetic energy of the tub and the work done by the frictional force. The tub keeps sliding until all of its kinetic energy is transferred to the work done by friction. The kinetic energy of the lighter tub K1 is transferred to the work done by friction W3, and the kinetic energy of the heavier tub K2 is transferred to W3. ∵ K1 > K2, ∴W3 > W4.

The work done by friction is calculated by multiplying friction with the distance an object travels. W3 = f1 × d1 and W4 = f2 × d2. By rearranging the equation, we can have: d1 =W3f1and d2 =W4f2. As mentioned before, W3 > W4 and f1 < f2, we get the final result, d1, > d2, the distance the lighter tub travels is longer than the distance the heavier tub travels.

Both of my hypothesis and the possible explanation are correct, but when I was thinking of that explanation I simply thought that the object that encounters greater friction will go a shorter distance than an object that encounters less friction. This explanation is not going to work if the heavier tub moves a lot faster at first or if it just have a lot more kinetic energy than the lighter tub out of some other reason. Now I know more detail about why the heavier object moves a shorter distance: The heavier tub has a lower kinetic energy when it leaves the rubber band and moves on its own due to its mass and friction, and its kinetic energy also transfers to work done by friction faster than the lighter tub due to the mass of the heavier tub. The thing that I did not realize before is that the second stage of the experiment is very important because it determines that heavier tubs will have less kinetic energy even before it moves on its own.

When I was making the hypothesis and thinking of a possible explanation, I used Newton’s second law F = ma and my logic. I thought that when the force applied is constant, the heavier the object, the less the acceleration and if the acceleration is big, it means that the object will reach a higher velocity and thus will travel further. However, I cannot prove it with math because perhaps the object reaches a high speed but travels only for a really short time.

Tubby the Tub Experiment

The result we got from the experiment is that as the total mass of the tub increases, the shorter the distance that it slides along the floor. Why is this so? Let’s assume that there are two tubs with different weights in them and find out the reason why the heavier tub travels a shorter distance than the lighter tub.

There are three stages in moving the tub with a rubber band. The first stage is when the rubber band hasn’t been released and is pulled back, the second stage when the rubber band is released and is going back to its original straight position, the third stage is when the rubber band is straight and the tub is going forward on its own.

We can use two methods to explain it, one with force and the other with energy. How does forces explain that the heavier tub travels a shorter distance?

Firstly, in the first stage, the rubber band has not been released, so there are no forces acting on the tubs. Then, the rubber band is released and is going back to its original straight position. Let’s assume that the force applied onto the tubs with the rubber band is F. As the force is applied with the same rubber band and the distance that it is pulled back is fixed, the force it applied will be the same no matter what the mass the tub in front of it is. Besides the force applied by the rubber band, there is also frictional force. The frictional force is calculated the equation F = μmg, which means that the mass of the tub is directly proportional to the friction between the tub and the floor. So the heavier tub is going to have more friction than the less heavier tub. Assume the friction of the heavier tub and the lighter tub be f1 and f2 respectively. The resultant force R1 applied on the heavier tub is (F-f1) N and the resultant force R2 applied on the lighter tub is (F-f2) N. As f1 > f2, R1 < R2. According to Newton’s Second Law of Motion, F=ma. The lighter tub has a bigger resultant force, so it is going to have a larger acceleration.

A larger acceleration means that the lighter tub is going to reach a higher velocity when it leaves the rubber band and begins to travel on its own, entering the second stage.in the second stage, there is only one force acting: the frictional force. As mentioned before, the formula of friction is F = μmg and Newton’s Second Law of Motion is F=ma. After substituting the second formula into the first, we get ma = μmg, and after rearranging it, we get a = μg. This means that the acceleration of the heavier and the lighter tub in the second stage does not relate to their mass, so they have the same acceleration slowing down. While their acceleration is the same, the velocity of the lighter tub is higher. The equation of acceleration is: a=(V_2- V_1)/∆t,by rearranging it we can get: t=V_(2-V_1 )/a. In this case, the value of V2 is 0, and the acceleration is a negative value, so t ∝ V, where t is the time the tubs need to reduce its speed to 0 and V is its current speed. As a result, the lighter tub will need to travel a longer time before it stops. The formula that connects the acceleration, time and distance together is d = 1/2at^2. In this case, the value of the heavier tub and the lighter tub is the same, so the longer the time it takes, the further the distance. Since the lighter tub needs a longer time to slow down, it will travel a longer distance than the heavier tub.

Another way to explain this is with energy.

In the first stage, the only energy is the elastic potential energy in the rubber bands. Elastic potential energy (measured in the unit joules) is equal to ½ multiplied by the stretch length ("x") squared, multiplied by the spring constant "k." As the stretch length of the rubber band in the experiment is a fixed value and the spring constant is also fixed as we are using the same rubber band, the elastic potential energy before the release of the rubber band is always the same no matter what the mass of the tub in front of it is. As a result, we can have this equation: P1 = P2, where P1 and P2 are the elastic potential energy of the rubber bands with tubs of different masses in front of them.

In the second stage, the rubber band is released and is going back to its original straight position. In this stage, the potential energy of the rubber bands is transferring into two energy: the kinetic energy of the tub and the work done by friction between the tub and the floor. How does the mass of the tub affect the work done by friction? The equation to calculate the work done by friction is f × d, the frictional force times the distance the object travels. The distance the tub, heavy or light, travels when the rubber band went from a pulled-back position to a straight line is a fixed value. The heavier of the two tubs is going to have more friction than the less heavy tub as it has more mass. As a result, the work done by the friction of the heavier tub is larger than of the lighter tub. To find the relationship between the kinetic energy of the two tubs, we can have the following equations: ∵ P1 = P2, ∴ K1 + W1 = K2 + W2, where K1 and W1 are the kinetic energy and the work done by the friction of the lighter tub and K2 and W2 are of the heavier tub. As W1 < W2, K1 > K2.

In the third stage, after the rubber band is straight and the tub is going forward on its own, there is no more potential energy transferring into kinetic energy, so the only forces left are the kinetic energy of the tub and the work done by the frictional force. The tub keeps sliding until all of its kinetic energy is transferred to the work done by friction. The kinetic energy of the lighter tub K1 is transferred to the work done by friction W3, and the kinetic energy of the heavier tub K2 is transferred to W3. ∵ K1 > K2, ∴W3 > W4.

The work done by friction is calculated by multiplying friction with the distance an object travels. W3 = f1 × d1 and W4 = f2 × d2. By rearranging the equation, we can have: d1 =W3f1and d2 =W4f2. As mentioned before, W3 > W4 and f1 < f2, we get the final result, d1, > d2, the distance the lighter tub travels is longer than the distance the heavier tub travels.

Both of my hypothesis and the possible explanation are correct, but when I was thinking of that explanation I simply thought that the object that encounters greater friction will go a shorter distance than an object that encounters less friction. This explanation is not going to work if the heavier tub moves a lot faster at first or if it just have a lot more kinetic energy than the lighter tub out of some other reason. Now I know more detail about why the heavier object moves a shorter distance: The heavier tub has a lower kinetic energy when it leaves the rubber band and moves on its own due to its mass and friction, and its kinetic energy also transfers to work done by friction faster than the lighter tub due to the mass of the heavier tub. The thing that I did not realize before is that the second stage of the experiment is very important because it determines that heavier tubs will have less kinetic energy even before it moves on its own.

When I was making the hypothesis and thinking of a possible explanation, I used Newton’s second law F = ma and my logic. I thought that when the force applied is constant, the heavier the object, the less the acceleration and if the acceleration is big, it means that the object will reach a higher velocity and thus will travel further. However, I cannot prove it with math because perhaps the object reaches a high speed but travels only for a really short time.