A ball is thrown with speed v₀ in a direction that forms an angle q with the horizontal. The range R is the ball's horizontal displacement from the start to the point at which the ball returns to the same horizontal level. Assume that the ball experiences no air resistance.

When the launch speed is doubled while the launch angle remains the same, the range

1. is halved because doubling the launch speed cuts the time of flight in half so that the ball can go only half as far
2. remains the same because doubling the launch speed cuts the time of flight in half, but the ball will go twice as fast; so half the time elapsed and double the speed results in the same range
3. is doubled; reason: when the launch speed is doubled the time of flight stays the same (the time to reach the top stays the same because the ball goes twice as fast and has twice as far to go up); thus, since the horizontal velocity component is doubled, the ball travels twice as far in the same time
4. is quadrupled; reason: when the launch speed is doubled both the horizontal and vertical velocity components are doubled; doubling the vertical component implies that the time to the top gets doubled because it takes twice as long to reduce this velocity component to zero; thus, the total travel time doubles since it is equal to twice the time to the top; double the travel time multiplied by double the horizontal velocity component gives four times the horizontal displacement