Freefall 自由下落 In the absence of frictional drag, an object near the surface of the earth will fall with the constant acceleration of gravity g. Position and speed at any time can be calculated from the motion equations. 在没有摩擦阻力的情况下，接近地球表面的物体将以恒定的重力加速度g下落。任意时刻的位置和速率可以通过运动方程计算。 Illustrated here is the situation where an object is released from rest. Its position and speed can be predicted for any time after that. Since all the quantities are directed downward, that direction is chosen as the positive direction in this case. 这里展示的是一个物体从静止释放的情况。其位置和速率可以在释放后任何时间被预测。由于所有量都是向下的，因此在此情况下选择向下为正方向。 At time t = s after being dropped, the speed is vy = m/s = ft/s , The distance from the starting point will be y = m= ft Enter data in any box and click outside the box. 在时间 t = s 后被释放，速度为 v_y = m/s = ft/s，从起点的距离为 y = m = ft，任选一个框输入数据并点击框外。 中文译文中的待填/计算数值依次对应：1：t 2：v 3：vc 4：y 5：yc。实际数值以上方原输入框为准。

Note that you can enter a distance (height) and click outside the box to calculate the freefall time and impact velocity in the absence of air friction. But the calculation assumes that the gravity acceleration is the surface value g = 9.8 m/s², so if the height is great enough for gravity to have changed significantly the results will be incorrect.

Horizontal Launch

All the parameters of a horizontal launch can be calculated with the motion equations, assuming a downward acceleration of gravity of 9.8 m/s².

水平发射的各个参数都可以用运动方程计算，假设重力加速度为9.8 m/s²。

Time of flight t = s Vertical impact velocity vy = m/s Launch velocity v0 = m/s Height of launch h = m Horizontal range R = m Calculation is initiated by clicking on the formula in the illustration for the quantity you wish to calculate. 通过点击示意图中你希望计算的量的公式来启动计算。 飞行时间 t = s 垂直撞击速度 v_y = m/s 发射速度 v_0 = m/s 发射高度 h = m 水平射程 R = m 中文译文中的待填/计算数值依次对应：1：t11 2：vy11 3：v011 4：h11 5：r11。实际数值以上方原输入框为准。

General Ballistic Trajectory

Will it clear the fence?

The basic motion equations can be solved simultaneously to express y in terms of x.

For launch velocity v0 = m/s = ft/s, launch angle θ = degrees, and horizontal range x = m = ft, the calculated height is y = m = ft. The time of flight is t = s. 对于初速度 v₀ = m/s = ft/s，发射角 θ = 度，水平射程 x = m = ft，计算出的最高高度为 y = m = ft。飞行时间是 t = 秒。 中文译文中的待填/计算数值依次对应：1：v0 2：v0c 3：a 4：x 5：xc 6：y 7：yc 8：t。实际数值以上方原输入框为准。

Where will it land?

Where will it land?

The basic motion equations give the position components x and y in terms of the time. Solving for the horizontal distance in terms of the height y is useful for calculating ranges in situations where the launch point is not at the same level as the landing point.

Launch velocity v0 = m/s = ft/s, launch angle θ = degrees, and trajectory height y = m = ft, 发射速度 v₀ = m/s = ft/s，发射角 θ = 度，轨迹高度 y = m = ft， 中文译文中的待填/计算数值依次对应：1：v0 2：v0c 3：a 4：y 5：yc。实际数值以上方原输入框为准。

The two calculated times are t1 = s and t2 = s. The corresponding ranges are x1 = m =ft and x2 = m =ft. 计算出的两个时间分别是 t₁ = s 和 t₂ = s。对应的射程分别是 x₁ = m = ft 和 x₂ = m = ft。 中文译文中的待填/计算数值依次对应：1：t1 2：t2 3：x1 4：x1c 5：x2 6：x2c。实际数值以上方原输入框为准。

Note that the value y in the illustration is downward and it is presumed that upward is positive. To reproduce the scenario in the diagram, the input value of y should be negative. 请注意，图中示例中的y值向下，且假设向上为正方向。要重现图中的情景，输入的y值应为负数。

Launch Velocity

The launch velocity of a projectile can be calculated from the range if the angle of launch is known. It can also be calculated if the maximum height and range are known, because the angle can be determined.

From the range relationship, the launch velocity can be calculated. For range R = m = ft, and launch angle θ = degrees, the launch velocity is v0 = m/s = ft/s. 从射程关系可以计算出发射速度。对于射程 R = m = ft，发射角 θ = degrees，发射速度为 v₀ = m/s = ft/s。 中文译文中的待填/计算数值依次对应：1：R 2：Rc 3：a 4：v 5：vc。实际数值以上方原输入框为准。

Launch Velocity

The launch velocity of a projectile can be calculated from the range if the angle of launch is known. It can also be calculated if the maximum height and range are known, because the angle can be determined.

For range
R = m = ft,
and peak height
h = m = ft,

the launch velocity is
v₀m/s = ft/s.

The required launch angle is
θ = degrees.

Angle of Launch

Variation of the launch angle of a projectile will change the range. If the launch velocity is known, the required angle of launch for a desired range can be calculated from the motion equations.

From the range relationship, the angle of launch can be determined. For range R = m = ft, and launch velocity v0 = m/s = ft/s. there are two solutions for the launch angle. θ1 = degrees, θ2 = degrees, 从射程关系可以确定发射角。对于射程 R = m = ft，发射速度 v₀ = m/s = ft/s，存在两个解的发射角。θ₁ = 度，θ₂ = 度， 中文译文中的待填/计算数值依次对应：1：R 2：Rc 3：v 4：vc 5：a1 6：a2。实际数值以上方原输入框为准。