Calculate the change in the kinetic energy (KE) of the bottle when the mass is increased. Use the formula [tex]$K E=\frac{1}{2} m v^2$[/tex], where [tex]$m$[/tex] is the mass and [tex]$v$[/tex] is the speed (velocity). Assume that the speed of the soda bottle falling from a height of 0.8 m will be [tex]$4 m / s$[/tex], and use this speed for each calculation. Record your calculations in Table A of your Student Guide.
When the mass of the bottle is 0.125 kg, the KE is
[ ] [tex]$kg \cdot m ^2 / s ^2$[/tex].
When the mass of the bottle is 0.250 kg, the KE is
[ ] [tex]$kg \cdot m ^2 / s ^2$[/tex]
When the mass of the bottle is 0.375 kg, the KE is
[ ] [tex]$kg \cdot m ^2 / s ^2$[/tex]
When the mass of the bottle is 0.500 kg, the KE is [tex]$kg \cdot m ^2 / s ^2$[/tex]
Answer (1)
Calculate the kinetic energy for mass m = 0.125 k g : K E = 2 1 ( 0.125 ) ( 4 2 ) = 1.0 k g ⋅ m 2 / s 2 .
Calculate the kinetic energy for mass m = 0.250 k g : K E = 2 1 ( 0.250 ) ( 4 2 ) = 2.0 k g ⋅ m 2 / s 2 .
Calculate the kinetic energy for mass m = 0.375 k g : K E = 2 1 ( 0.375 ) ( 4 2 ) = 3.0 k g ⋅ m 2 / s 2 .
Calculate the kinetic energy for mass m = 0.500 k g : K E = 2 1 ( 0.500 ) ( 4 2 ) = 4.0 k g ⋅ m 2 / s 2 .
1.0 , 2.0 , 3.0 , 4.0
Explanation
Understanding the Problem We are asked to calculate the kinetic energy (KE) of a soda bottle for different masses, given the formula K E = f r a c 1 2 m v 2 , where m is the mass and v is the velocity. We are given that the velocity v is constant at 4 m / s . We need to calculate the KE for four different masses: 0.125 kg, 0.250 kg, 0.375 kg, and 0.500 kg.
Calculating KE for m = 0.125 kg First, let's calculate the KE when the mass m = 0.125 k g . Using the formula, we have: K E = f r a c 1 2 m v 2 = f r a c 1 2 ( 0.125 k g ) ( 4 m / s ) 2 = f r a c 1 2 ( 0.125 ) ( 16 ) = 0.125 t im es 8 = 1.0 k g c d o t m 2 / s 2
Calculating KE for m = 0.250 kg Next, let's calculate the KE when the mass m = 0.250 k g . Using the formula, we have: K E = f r a c 1 2 m v 2 = f r a c 1 2 ( 0.250 k g ) ( 4 m / s ) 2 = f r a c 1 2 ( 0.250 ) ( 16 ) = 0.250 t im es 8 = 2.0 k g c d o t m 2 / s 2
Calculating KE for m = 0.375 kg Now, let's calculate the KE when the mass m = 0.375 k g . Using the formula, we have: K E = f r a c 1 2 m v 2 = f r a c 1 2 ( 0.375 k g ) ( 4 m / s ) 2 = f r a c 1 2 ( 0.375 ) ( 16 ) = 0.375 t im es 8 = 3.0 k g c d o t m 2 / s 2
Calculating KE for m = 0.500 kg Finally, let's calculate the KE when the mass m = 0.500 k g . Using the formula, we have: K E = f r a c 1 2 m v 2 = f r a c 1 2 ( 0.500 k g ) ( 4 m / s ) 2 = f r a c 1 2 ( 0.500 ) ( 16 ) = 0.500 t im es 8 = 4.0 k g c d o t m 2 / s 2
Final Answer Therefore, when the mass of the bottle is 0.125 kg, the KE is 1.0 k g c d o t m 2 / s 2 . When the mass of the bottle is 0.250 kg, the KE is 2.0 k g c d o t m 2 / s 2 . When the mass of the bottle is 0.375 kg, the KE is 3.0 k g c d o t m 2 / s 2 . When the mass of the bottle is 0.500 kg, the KE is 4.0 k g c d o t m 2 / s 2 .
Examples
Understanding kinetic energy is crucial in many real-world scenarios. For instance, when designing vehicles, engineers must consider the kinetic energy involved in collisions to improve safety features like airbags and crumple zones. Similarly, in sports, understanding kinetic energy helps athletes optimize their performance, such as maximizing the energy transferred to a ball during a swing or kick. Even in amusement park rides, the principles of kinetic energy are applied to create thrilling yet safe experiences.
