Nate the Skate was an avid physics student whose main non-physics interest in life was high-speed skateboarding. In particular, Nate would often don a protective suit of Bounce-Tex, which he invented, and after working up a high speed on his skateboard, would collide with some object. In this way, he got a gut feel for the physical properties of collisions and succeeded in combining his two passions.* On one occasion, the Skate, with a mass of 121 kg, including his armor, hurled himself against a 839-kg stationary statue of Isaac Newton in a perfectly elastic linear collision. As a result, Isaac started moving at 1.57 m/s and Nate bounced backward. What were Nate\'s speeds immediately before and after the collision? (Enter positive numbers.) Ignore friction with the ground.
Before:?
After:?
m1v1i = m1v1f + m2v2f.
Pay attention to signs
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.
Let's denote Nate's velocity before the collision as V1 and his velocity after the collision as V2.
According to the principle of conservation of momentum:
Total momentum before collision = Total momentum after collision
The total momentum before the collision is the product of Nate's mass (121 kg) and his initial velocity (V1). The total momentum after the collision is the product of Nate's mass (121 kg) and his final velocity (V2), plus the product of Isaac Newton's mass (839 kg) and his final velocity (1.57 m/s).
Therefore:
121 kg * V1 = 121 kg * V2 + 839 kg * 1.57 m/s
Now, let's apply the principle of conservation of kinetic energy:
Total kinetic energy before collision = Total kinetic energy after collision
The total kinetic energy before the collision is given by the equation: (1/2) * Nate's mass * V1^2. The total kinetic energy after the collision is given by the equation: (1/2) * Nate's mass * V2^2 + (1/2) * Isaac Newton's mass * (1.57 m/s)^2.
Therefore:
(1/2) * 121 kg * V1^2 = (1/2) * 121 kg * V2^2 + (1/2) * 839 kg * (1.57 m/s)^2
Now let's solve these two equations simultaneously to find the values of V1 and V2.
Note: Since we are only interested in the magnitude of the velocities, we can ignore the negative signs.
Here are the steps to solve for V1 and V2:
1. Solve the first equation for V2:
121 kg * V1 = 121 kg * V2 + 839 kg * 1.57 m/s
Rearranging the equation:
V2 = (121 kg * V1 - 839 kg * 1.57 m/s) / 121 kg
2. Substitute the value of V2 in the second equation:
(1/2) * 121 kg * V1^2 = (1/2) * 121 kg * ( (121 kg * V1 - 839 kg * 1.57 m/s) / 121 kg )^2 + (1/2) * 839 kg * (1.57 m/s)^2
3. Simplify the equation and solve for V1:
605 V1^2 = 252142.47 + 5240.271803 V1 - 90.217135 V1^2
Combine like terms:
695 V1^2 - 5240.271803 V1 - 252142.47 = 0
4. Solve the quadratic equation for V1 using the quadratic formula:
V1 = (-(-5240.271803) ± √((-5240.271803)^2 - 4 * 695 * (-252142.47))) / (2 * 695)
Calculate the values inside the square root:
V1 = (5240.271803 ± √(27433697.682981 - (-1,756,361,805.435)))) / 1390
V1 = (5240.271803 ± √1,783,695,103.682981) / 1390
V1 = (5240.271803 ± 42,238.60493) / 1390
5. Calculate the two possible values for V1 and find the corresponding values of V2 using the first equation:
V1 = (5240.271803 + 42,238.60493) / 1390 = 44.6 m/s
V2 = (121 kg * 44.6 m/s - 839 kg * 1.57 m/s) / 121 kg = 3.41 m/s
V1 = (5240.271803 - 42,238.60493) / 1390 = -25.7 m/s (ignore the negative sign)
V2 = (121 kg * (-25.7 m/s) - 839 kg * 1.57 m/s) / 121 kg = -4.085 m/s (ignore the negative sign)
The velocities immediately before and after the collision are approximately:
Before: V1 = 44.6 m/s
After: V2 = 3.41 m/s
Therefore, Nate's speed immediately before the collision was 44.6 m/s, and his speed immediately after the collision was 3.41 m/s.
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of momentum:
In a perfectly elastic collision, the total momentum before and after the collision remains the same.
Before the collision, the total momentum is given by:
P_before = m1 * v1 + m2 * v2,
where m1 and v1 are the mass and velocity of Nate, and m2 and v2 are the mass and velocity of the statue of Isaac Newton.
After the collision, the total momentum is given by:
P_after = m1 * v1' + m2 * v2',
where v1' and v2' are the velocities of Nate and Isaac Newton after the collision.
Since we know the mass of Nate (m1 = 121 kg), the mass of the statue (m2 = 839 kg), and the velocity of Isaac Newton after the collision (v2 = 1.57 m/s), we can use the conservation of momentum to find Nate's velocity before the collision.
2. Conservation of kinetic energy:
In a perfectly elastic collision, the total kinetic energy remains the same before and after the collision.
Before the collision, the total kinetic energy is given by:
KE_before = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2,
After the collision, the total kinetic energy is given by:
KE_after = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2.
Since we know the mass of Nate (m1 = 121 kg), the mass of the statue (m2 = 839 kg), and the velocity of Isaac Newton after the collision (v2 = 1.57 m/s), we can use the conservation of kinetic energy to find Nate's velocity after the collision.
By solving these two equations simultaneously, we can find Nate's velocities before and after the collision.
Now, let's solve the problem using the given information:
1. Conservation of momentum:
The total momentum before the collision is equal to the total momentum after the collision:
m1 * v1 + m2 * v2 = m1 * v1' + m2 * v2'.
Substituting the given values:
121 kg * v1 + 839 kg * 0 m/s = 121 kg * v1' + 839 kg * 1.57 m/s.
2. Conservation of kinetic energy:
The total kinetic energy before the collision is equal to the total kinetic energy after the collision:
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * m1 * v1'^2 + (1/2) * m2 * v2'^2.
Substituting the given values:
(1/2) * 121 kg * v1^2 + (1/2) * 839 kg * 0 m/s + (1/2) * 121 kg * v1'^2 + (1/2) * 839 kg * 1.57 m/s = (1/2) * 121 kg * v1'^2 + (1/2) * 839 kg * 0 m/s + (1/2) * 121 kg * v1'^2 + (1/2) * 839 kg * 1.57 m/s.
Simplifying both equations, we have a system of two equations with two unknowns (v1 and v1').
Solving these equations will give us Nate's velocities before and after the collision.