What minimum speed does a 170g puck need to make it to the top of a 5.0m long 28 degree frictionless ramp?

To find the minimum speed required for the puck to reach the top of the ramp, we can use energy conservation principles. The gravitational potential energy at the top of the ramp (mgh) must equal the initial kinetic energy (0.5mv²) at the bottom. The height ( h ) can be calculated as ( h = L \sin(\theta) ), where ( L = 5.0 ) m and ( \theta = 28^\circ ).

Calculating ( h ):

[ h = 5.0 \sin(28^\circ) \approx 5.0 \times 0.469 = 2.345 , \text{m} ]

Thus, the potential energy at the top is:

[ PE = mgh = 0.170 \times 9.81 \times 2.345 \approx 3.930 , \text{J} ]

Setting this equal to the kinetic energy gives:

[ 0.5mv^2 = 3.930 \implies 0.5 \times 0.170 \times v^2 = 3.930 \implies v^2 = \frac{3.930}{0.085} \approx 46.24 \implies v \approx 6.80 , \text{m/s} ]

Therefore, the minimum speed needed is approximately 6.80 m/s.