The spring of a spring gun has force constant k = 400 N/m and negligible mass. The spring is compressed 6.00 cm, and a ball with mass 0.0300 kg is placed in the horizontal barrel against the compressed spring. The spring is then released, and the ball is propelled out the barrel of the gun. The barrel is 6.00 cm long, so the ball leaves the barrel at the same point that it loses contact with the spring. The gun is held so that the barrel is horizontal. (a) Calculate the speed with which the ball leaves the barrel if you can ignore friction. (b) Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 N acts on the ball as it moves along the barrel. (c) For the situation in part (b), at what position along the barrel does the ball have the greatest speed, and what is that speed? (In this case, the maximum speed does not occur at the end of the barrel.)
Solution 80P Step 1 of 7: (a) Calculate the speed with which the ball leaves the barrel if you can ignore friction. In the given case, the potential energy stored in the string when it is compressed will be transferred as the kinetic energy for the ball when the spring is released. Using this logic, we need to calculate the speed of the ball when it leaves the barrel.(neglecting frictional force) Given data, Spring constant or force constant, k= 400 N/m Compression length, x= 6 cm = 0.06 m Mass of the ball, m= 0.03 kg To find, Speed with which ball leaves the barrel, v= Step 2 of 7: Potential Energy stored in spring= Kinetic energy of ball 1 1 2kx = m2 2 Solving for speed v, 2 v = km k v = x m Substituting x= 0.06 m , k= 400 N/m and m=0.03 kg (400 N/m ) v = (0.06 m) (0.03 kg) v = 6.93 m/s Therefore, the speed of the ball when it leaves the barrel is 6.93 m/s. Step 3 of 7: (b) Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 N acts on the ball as it moves along the barrel. The spring potential energy will be suppressed by some amount by the restoring force (F) that is due to friction and the remaining energy will be transferred as kinetic energy to the ball. Since in this case, we are considering the effect of friction on the ball The equation changes to Potential energy stored in the spring- frictional force work done = kinetic energy of ball 1 2 1 2 2kx Fx = mv2 Solving for speed v, kx 2Fx v = m Step 4 of 7: Substituting x= 0.06 m , k= 400 N/m, F= 6N and m=0.03 kg (400 N/m)(0.06 m) 2(6N)(0.06 m) v = (0.03 kg) v = (0.03 kg) v = 24 v= 4.9 m/s Therefore, when friction force is taken into account; the speed of the ball when it leaves the barrel is 4.9 m/s.