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As a ball falls, it speeds up and drops lower — gaining kinetic energy while losing potential energy. Does the total stay fixed?
- Mechanical energy = kinetic energy + potential energy. For a ball dropped from height h : at the top, KE = 0 , PE = mgh , so mechanical energy = mgh .
- After falling for time t , the lost potential energy 1 2 mg 2 t 2 exactly equals the gained kinetic energy 1 2 mg 2 t 2 , so the total stays mgh .
- Conservation of mechanical energy: if no other external forces act, the mechanical energy of an object stays constant as it moves under gravity.
In this Activity, we will release a pendulum bob from a marked height and watch whether it swings back up to almost the same level, to test conservation of mechanical energy.
- Set up a simple pendulum. On paper behind it, draw a horizontal line at the level of the resting bob's starting height.
- Take the bob to one side to point P at that line and release it. Watch the extreme points of the first few swings.
- At P the bob has only potential energy ( mgh ); at the bottom Q only kinetic energy; at R on the far side only potential energy again. The bob reaches almost the same height — mechanical energy is conserved.
- In real life it slowly stops, because friction at the support and air resistance remove energy.
- Conservation of mechanical energy — if no external forces act, the total mechanical energy (kinetic + potential) of an object remains constant.
- Solving problems directly with Newton's laws can get cumbersome. By keeping track of total mechanical energy, we can often find an object's final speed or position without working through every step of the motion.
What is the speed of a child at the bottom of a slide of height h?
PE at top = mgh converts to KE at bottom = 1 2 mv 2 (ignoring friction):
1
2
mv
2
= mgh
⇒ v
=
2gh
.
The speed depends only on the height h — not on the shape of the slide or the child's mass.
A 10000 kg truck at 72 km h⁻¹ runs onto a 30° sand ramp (sand force 50000 N). Minimum ramp length to stop it? g = 10, and the truck rises 1 m for every 2 m along the ramp.
v
= 72
km h
-1
= 20
m s
-1
. Initial KE
=
1
2
× 10000 × 20
2
= 2000000
J
.
For ramp length
d
: height gained
= d/2
, so PE gained
= mg(d/2)
= 10000×10×(d/2)
= 50000 d
. Work by sand
= -50000 d
.
Work-energy:
-50000 d
= (50000 d) - 2000000
⇒ 2000000
= 100000 d
⇒ d
= 20
m
.
- Mechanical energy is just one part of a bigger picture. In nature, the total energy of an object or system not acted on by external forces stays constant — energy only changes form.
- 7. Mechanical energy of the ball just before it hits the ground: at the ground h=0 so PE=0 , and KE = 1 2 mv 2 = mgh (since v 2 = 2gh ). So mechanical energy = mgh — the same as at the top.
- 8. Roller-coaster ball (A, B, C…): at high points PE is large and KE small; at low points KE is large and PE small. Each later peak (C, D, E) is lower because some mechanical energy is lost to friction (and air resistance) along the track.
NCERT Question 3 — When a ball thrown upwards
When a ball thrown upwards reaches its highest point, tick which of the following statement(s) are correct?
(i) The force acting on the ball is zero.
(ii) The acceleration of the ball is zero.
(iii) Its kinetic energy is zero.
(iv) Its potential energy is maximum.
NCERT Question 14 — The potential energy-displacement graph of
The potential energy-displacement graph of a 0.5 kg ball moving along a frictionless track is shown in Fig. 7.39. At O, the velocity of the ball is 0 m s⁻¹ and potential energy is 30 J. Calculate the velocity of the ball at P, Q and R.
NCERT Question 15 — A coconut of mass 1.5
A coconut of mass 1.5 kg falls from the top of a coconut tree onto the wet sand on a beach. The height of the tree is 10 m. On impact, the coconut comes to rest by making a depression in the sand. (i) Calculate the velocity of the coconut just before it hits the sand. (ii) Assume that the average resistive force of sand is 3000 N and all of the coconut's energy is used to create the depression in the sand. Calculate the depth of the depression the coconut makes in the sand. Assume g = 10 m s⁻².
-
Write the expressions for kinetic energy and for gravitational potential energy.
Show Answer
K = 1 2 mv 2 and U = mgh . Both are measured in joule (J). -
If the velocity of a vehicle is doubled, what happens to its kinetic energy?
Show Answer
It becomes four times larger, because K depends on v 2 . -
What is the mechanical energy of an object, and when is it conserved?
Show Answer
The sum of its kinetic and potential energy. It is conserved when no other external force (such as friction) acts on the object. -
A pendulum bob swings from P to Q to R. Where is its kinetic energy the largest?
Show Answer
At Q, the lowest point — there the potential energy is zero and all of it has become kinetic energy.
