Energy and Power

(Assume g = 10 m/s² or 10 N/kg unless otherwise stated)

1. An object A of mass 10 kg is moving with a velocity of 6 m/s. Calculate its kinetic energy and its momentum. If a constant opposing force of 20 N suddenly acts on A, find the times it takes to come to rest and the distance through which it moves.  [Answer: 180 J, 60 N s, 3 s, 9 m]

2. (a) A car of mass 1000 kg moving on a horizontal road with a steady velocity of 10 m/s has a total frictional force on it of 400 N. Find the power due to the engine. (b) The car now climbs a hill at an angle of 8 degree to the horizontal. Assuming the frictional force stays constant at 400 N, what engine power in now needed to keep the car moving at 10 m/s? [Answer: (a) 4 kW (b) 17.9 kW]

3. Sand drops vertically at the rate of 2 kg/s onto a conveyor belt moving horizontally with a velocity of 0.1 m/s. Calculate (i) the extra power needed to keep the belt moving, (ii) the rate of change of kinetic energy of the sand. Why is the power twice as great as the rate of change of kinetic energy? [Answer: (i) 0.02 W (ii) 0.01 W]

4. A ball of mass 0.1 kg is thrown vertically upwards with an initial speed of 20 m/s. Calculate (i) the time taken to return to the thrower, (ii) the maximum height reached (iii) the kinetic and potential energies of the ball half-way up. [Answer: (i) 4 s (ii) 20 m (iii) 10 J, 10 J ]

5. A 4 kg ball moving with a velocity of 10.0 m/s collides with a 16 kg ball moving with a velocity of 4.0 m/s (i) in the same direction (ii) in the opposite direction. Calculate the velocity of the balls in each case if they coalesce on impact, and the loss of energy resulting from the impact. State the principle used to calculate the velocity. [Answer: (i) 5.2 m/s, 58 J (ii) 1.2 m/s, 314 J ]

6. A ball of mass 0.1 kg is thrown vertically upward with a velocity of 20 m/s. What is the potential energy at the maximum height? What is the potential energy of the ball when it reaches three-quarters of the maximum height will moving upward? [Answer: 20 J, 15 J ]

7. A stationary mass explodes into two parts of mass 4 units and 40 units respectively. If the larger mass has an initial kinetic energy of 100 J, what is the initial kinetic energy of the smaller mass? Explain your calculation. [Answer: 1000 J]

8. A car of mass 1000 kg moves at a constant speed of 20 m/s along a horizontal road where the frictional force is 200 N. Calculate the power developed by the engine. If the car moves up an incline at the same constant speed, calculate the new power developed by the engine. Assume that the frictional force is still 200 N and that sin Ø = 1/20, where Ø is the angle of the incline to the horizontal. [Answer: 4 kW, 14 kW]

9. A horizontal force of 2000 N is applied to a vehicle of mass 400 kg which is initially at rest on a horizontal surface. If the total force opposing motion is constant at 800 N, calculate (i) the acceleration of the vehicle (ii) the kinetic energy of the vehicle 5s after the force is first applied (iii) the total power developed 5s after the force is first applied. [Answer: (i) 3 m/s² (ii) 45000 J (iii) 30 kW]

10. A railway truck of mass 4 x 10⁴ kg moving at a velocity of 3 m/s collides with another truck of mass 2 x10⁴ kg which is at rest. The couplings join and the trucks move off together. What fraction of the first truck's initial kinetic energy remains as kinetic energy of the two trucks after the collision? Is energy reserved in a collision such as this? Explain your answer briefly. [Answer: 2/3]

11. An - particle having a speed of 1.00 x 10⁶ m/s collides with a stationary proton which gain an initial speed of 1.60 x 10⁶ m/s in the direction in which the alpha - particle was traveling. What is the speed of the alpha - particle immediately after the collision? How much kinetic energy is gained by the proton in the collision? It is known that this collision is perfectly elastic. Explain what this means. (Mass of  alpha - particle = 6.64 x 10^-27 kg, mass of proton = 1.66 x 10^-27 kg) [Answer: 6 x 10⁵ m/s, 2.1 x 10^-15 J]

12. (a) A particle of mass m, initially at rest, is acted upon by a constant force until its velocity is v. Show that the kinetic energy of the particle is 1/2mv². (b) A train of mass 2.0 x 10⁵ kg moves at a constant speed of 72 km/h up a straight incline against a frictional force of 1.28 x 10⁴ N. The incline is such that the train rises vertically 1.0 m for every 100 m traveled along the incline. Calculate (i) the rate of increase per second of the potential energy of the train, (ii) the necessary power developed by the train. [Answer: (b) (i) 400 kW (ii) 656 kW]

13. Two trolleys P and Q of masses 0.50 kg and 0.30 kg respectively are held together on a horizontal track against a spring whish is in a state of compression. When the spring is released the trolleys separate freely and P moves to the left with an initial velocity of 6 m/s. Calculate (a) the initial velocity of Q (b) the initial total kinetic energy of the system. Calculate also the initial velocity of Q if trolley P is held still when the spring under the same compression as before is released. [Answer: (a) 10 m/s (b) 24 J, 12.65 m/s]

14. Define linear momentum and state the principle of conservation of linear momentum. Explain briefly how you would attempt to verify this principle by experiment. Sand is deposited at a uniform rate of 20 kg per second and with negligible kinetic energy onto an empty conveyor belt moving horizontally at a constant speed of 10 meters per minute. Find (a) the force required to maintain constant velocity, (b) the power required to maintain constant velocity, and (c) the rate of change of kinetic energy of the moving sand. Why are the latter two quantities unequal? [Answer: (a) 10/3 N (b) 5/9 W (c) 5/18 W]