Prove the well‐known result that, for a given launch speed v0, the launch angle θ = 45° yields the maximum horizontal range R. Determine the maximum range. (Note that this result does not hold when aerodynamic drag is included in the analysis.)

Abug is 12 cm from the center of a turntable that is rotating with a frequency of 45 rev/min . what minimum coefficient friction is required so that the bug stays on the turntable?

You are driving directly behind a pickup truck, going at the same speed as the truck. a crate falls from the bed of the truck to the road. (a) will your car hit the crate before the crate hits the road if you neither brake nor swerve? (b) during the fall, is the horizontal speed of the crate more than, less than, or the same as that of the truck?

Aspaceship starting from a resting position accelerates at a constant rate of 9.8 meters per second per second. how far will the spaceship travel if its final speed is 1 percent the speed of light (300,000,000 m/s)? a. 3.1 × 106 m b. 2.94 × 108 m c. 7.8 × 1010 m d. 4.6 × 1013 m