a. 0.41 m
b. 5.72 m/s
c. i. For part (a), I chose the hand as the reference level since the penny was thrown from the hand and the height of the penny at the hand is zero and also, it is easier to calculate from a zero reference level.
ii. For part (b), I chose the ground as the reference level since the height of the penny above the ground is positive and the height of the penny when the penny hits the ground is zero and also, it is easier to calculate from a zero reference level.
d. 5.72 m/s
Explanation:
a. Use energy to find how high the penny goes above your hand before stopping.
Taking the hand as the ground level, and from the law of conservation of energy, the total mechanical energy at the hand, E equals the total mechanical energy when the penny stops in the air, E'.
E = E'
U + K = U' + K' where U = initial potential energy at hand level = mgh where h = height at hand level = 0, K = initial kinetic energy at hand level = 1/2mv² where v = speed at hand level = 2.85 m/s, U' = final potential energy at stopping level = mgh' where h' = height at stopping level, K = final kinetic energy at stopping level = 1/2mv'² where v = speed at stopping level = 0 m/s (since the penny momentarily stops)
So, U + K = U' + K'
mgh + 1/2mv² = mgh' + 1/2mv'²
substituting the values of the variables into the equation, we have
mg(0) + 1/2m(2.85 m/s)² = mgh' + 1/2m(0 m/s)²
0 + 1/2m(8.1225 m²/s²) = mgh' + 0
m(4.06125 m²/s²) = mgh'
h' = 4.06125 m²/s² ÷ g
h' = 4.06125 m²/s² ÷ 9.8 m/s²
h' = 0.41 m
(b) The penny then falls to the floor, 1.26 m below your hand. Use energy to find its speed just before it hits the floor.
Taking the hand as the ground level, and from the law of conservation of energy, the total mechanical energy when the penny stops in the air, E' equals the total mechanical energy on the ground, E"
E' = E"
U' + K' = U" + K" where U' = initial potential energy at stopping level = mgh" where h' = height at stopping level = height of penny above hand, h' + height of hand above ground = 0.41 m + 1.26 m = 1.67 m, K = initial kinetic energy at stopping level = 1/2mv'² where v = speed at stopping level = 0 m/s (since the penny momentarily stops), U = final potential energy at ground level = mgh₁ where h₁ = height at ground level = 0, K = final kinetic energy at ground level = 1/2mv"² where v" = speed at ground level,
So, U' + K' = U' + K'
mgh" + 1/2mv'² = mgh₁ + 1/2mv"²
substituting the values of the variables into the equation, we have
mg(1.67 m) + 1/2m(0 m/s)² = mg(0) + 1/2mv"²
1.67mg + 0 = 0 + 1/2mv"²
1.67mg = 1/2mv"²
1.67g = 1/2v"²
v"² = 2(1.67g)
v" = √[2(1.67g)]
v" = √[2(1.67 m × 9.8 m/s²)]
v" = √[2(16.366 m²/s²)]
v" = √[32.732 m²/s²)]
v" = 5.72 m/s
(c) Explain your choice of reference level for parts (a) and (b).
i. For part (a), I chose the hand as the reference level since the penny was thrown from the hand and the height of the penny at the hand is zero and also, it is easier to calculate from a zero reference level.
ii. For part (b), I chose the ground as the reference level since the height of the penny above the ground is positive and the height of the penny when the penny hits the ground is zero and also, it is easier to calculate from a zero reference level.
(d) Choose a different reference level and repeat part (b)
Taking the hand as the ground level, and from the law of conservation of energy, the total mechanical energy when the penny stops in the air, E' equals the total mechanical energy on the ground, E"
E' = E"
U' + K' = U" + K" where U' = initial potential energy at stopping level = mgh' where h' = height at stopping level = 0.41 m, K = initial kinetic energy at stopping level = 1/2mv'² where v' = speed at stopping level = 0 m/s (since the penny momentarily stops), U = final potential energy at ground level = mgh₁ where h₂ = height of hand above the ground level = height of ground below hand = -1.26 m(it is negative since the ground is below the hand), K = final kinetic energy at ground level = 1/2mv"² where v = speed at ground level,
So, U' + K' = U' + K'
mgh' + 1/2mv'² = mgh₂ + 1/2mv"²
substituting the values of the variables into the equation, we have
mg(0.41 m) + 1/2m(0 m/s)² = mg(-1.26 m) + 1/2mv"²
0.41mg + 0 = -1.26 mg + 1/2mv"²
0.41mg + 1.26mg = 1/2mv"²
1.67mg = 1/2mv"²
1.67g = 1/2v"²
v"² = 2(1.67g)
v" = √[2(1.67g)]
v" = √[2(1.67 m × 9.8 m/s²)]
v" = √[2(16.366 m²/s²)]
v" = √[32.732 m²/s²)]
v" = 5.72 m/s
