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A moving bicycle, a rolling ball — all moving things can do work. How much energy does motion carry, and how does it depend on speed?
- The energy an object has due to its motion is its kinetic energy . A motionless object has zero kinetic energy.
- For a constant force on mass \(m\), using \(v^2 = u^2 + 2as\) and \(W = F\times s = ma\times s\): $$W = \tfrac{1}{2}m(v^2 - u^2)$$
- By the work-energy theorem this is the change in energy. Starting from rest (\(u = 0\)), the kinetic energy of an object of mass \(m\) moving at velocity \(v\) is $$K = \tfrac{1}{2}mv^2$$
- KE has no direction and its unit is the joule (J). It increases with positive work (speeding up) and decreases with negative work (slowing down).
- Kinetic energy — the energy possessed by an object due to its motion; for mass m at velocity v, K = ½mv².
If a vehicle's velocity doubles, how does its kinetic energy change?
Initial \(K = \tfrac{1}{2}mv^2\). At \(2v\): \(K' = \tfrac{1}{2}m(2v)^2 = 4 \times \tfrac{1}{2}mv^2\).
The kinetic energy becomes 4 times the original value.
A cricket ball of mass 0.2 kg is bowled at about 154.8 km h⁻¹. Find its kinetic energy.
\(v = 154.8\ \text{km h}^{-1} = 43\ \text{m s}^{-1}\).
\( K = \tfrac{1}{2}mv^2 = \tfrac{1}{2} \times 0.2 \times 43^2 = 184.9\ \text{J} \).
A 15000 kg jet is stopped within 100 m by a wire exerting 367500 N. Find its velocity just before the hook caught.
Change in KE = work done by the wire (force opposes motion, so work is negative):
\( 0 - \tfrac{1}{2}\times 15000 \times v^2 = -(367500 \times 100) \)
\( v^2 = \dfrac{367500 \times 2 \times 100}{15000} = 4900 \Rightarrow v = 70\ \text{m s}^{-1} = 252\ \text{km h}^{-1} \) towards the carrier.
- 4. A (mass m) and B (mass 4m) have equal KE — ratio of velocities? \(\tfrac{1}{2}m v_A^2 = \tfrac{1}{2}(4m)v_B^2 \Rightarrow v_A^2 = 4 v_B^2 \Rightarrow v_A : v_B = 2 : 1\).
- 5. Does KE of an object at constant velocity change with position? No — kinetic energy depends only on mass and speed, so at constant velocity it stays the same wherever the object is.
NCERT Question 8 — A man of mass 60
A 60 kg man on a 100 kg scooter reaches velocity v; next day a 40 kg son joins. Same speed in same time — ratio of fuel used on the two days.
View the answer →NCERT Question 11 — A 10.0 kg block is
A 10 kg block with KE 180 J at 0 m has a variable force applied (Fig. 7.37). Find its speed at 0 m and 4 m; any negative acceleration?
View the answer →NCERT Question 13 — A 1000 kg car is
A 1000 kg car brakes to a stop (Fig. 7.38 speed-time). Describe A–B; KE at A; work by brakes B–C; what the KE becomes.
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