Conservation of mechanical energy on curve surface – problems and solutions

1. If ball’s speed at point A is 6 m/s, ball’s speed at point B is 92 m/s, and acceleration due to gravity is g = 10 m/s2. What is the height of point B (h)?

Known :

Speed of ball at point A (vA) = 6 m/sConservation of mechanical energy on curve surface – problems and solutions 1

Speed of ball at point B (vB) = 92 m/s

Acceleration due to gravity (g) = 10 m/s2

The height of A (hA) = 5.6 meters

The height of B (hB) = h

Wanted : The height of point B (h)

Solution :

The initial mechanical energy = the gravitational potential energy

At point A, ball’s speed = 0, so the kinetic energy of the ball = 0. KE = 1/2 m v2 = 1/2 m (0) = 0.

Ball at the height of 5.6 meters so the ball has the gravitational potential energy. The gravitational potential energy : PE = m g h = m (10)(5.6) = 56 m

The initial mechanical energy = the gravitational potential energy + kinetic energy = 56 m + 0 = 56 m

The final mechanical energy = the gravitational potential energy + kinetic energy

At point B, the height of ball is h. The gravitational potential energy : PE = m g h = m (10) h = 10 m h

The kinetic energy of the ball : KE = 1/2 m v2 = 1/2 m (92)2 = 1/2 m (92) = 46 m

The final mechanical energy = the gravitational potential energy + the kinetic energy = 10 m h + 46 m = m (10 h + 46)

The principle of conservation of mechanical energy :

The initial mechanical energy = the final mechanical energy

56 m = m (10 h + 46)

56 = 10 h + 46

56 – 46 = 10 h

10 = 10 h

h = 10/10

h = 1 meter

2.

If the initial velocity = 0, the acceleration due to gravity = 10 m/s2, then what is the speed at the height of B.

Known :Conservation of mechanical energy on curve surface – problems and solutions 2

The initial speed (vo) = 0

The initial height (ho) = 50 m – 10 m = 40 m

The final height (ht) = 0

Acceleration due to gravity (g) = 10 m.s-2

Wanted : The final speed (vt)

Solution :

The change of the kinetic energy :

ΔKE = 1/2 m (vt2 – vo2) = 1/2 m (vt2) = 1/2 m vt2

The change of the potential energy :

ΔPE = m g (ht – ho) = m (10)(0-40) = m (10)(-40) = – 400 m

Calculate the final speed (vt) using the equation of the principle of conservation of mechanical energy :

0 = ΔKE + ΔPE

0 = 1/2 m vt2 – 400 m

1/2 m vt2 = 400 m

1/2 vt2 = 400

vt2 = 2(400)

vt = √(400)(2)

vt = 20√2 m/s

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