Edexcel A Level Physics: Projectile Motion | 爱德思 A Level 物理：抛体运动

📚 Edexcel A Level Physics: Projectile Motion | 爱德思 A Level 物理：抛体运动

Projectile motion is a core topic in Edexcel A Level Physics mechanics. It combines constant horizontal velocity with uniformly accelerated vertical motion under gravity. A projectile is an object that moves under the force of gravity alone, so air resistance is usually ignored in the ideal model. Understanding projectile motion helps solve problems in sports, ballistics, engineering and many real-world applications.

抛体运动是爱德思 A Level 物理力学部分的核心内容。它将水平方向的匀速运动与竖直方向在重力作用下的匀加速运动结合在一起。抛体是仅在重力作用下运动的物体，因此理想模型通常忽略空气阻力。理解抛体运动有助于解决体育运动、弹道、工程以及许多现实世界中的问题。

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1. Scalar and Vector Quantities | 标量与矢量

Before analysing projectile motion, you must be confident with scalar and vector quantities. Displacement, velocity, acceleration and force are vectors; distance, speed and energy are scalars. In projectile problems, the initial velocity is a vector, and it must be resolved into horizontal and vertical components. These two components are independent of each other.

在分析抛体运动之前，你必须熟练掌握标量和矢量。位移、速度、加速度和力都是矢量；路程、速率和能量是标量。在抛体问题中，初速度是一个矢量，必须分解为水平分量和竖直分量。这两个分量彼此独立。

uₓ = u cos θ, u_y = u sin θ

Here u is the initial speed, θ is the launch angle to the horizontal, uₓ is the horizontal component and u_y is the vertical component. Students often forget that the launch angle is measured from the horizontal, not the vertical.

这里的 u 是初速度大小，θ 是相对水平方向的抛射角，uₓ 是水平分量，u_y 是竖直分量。学生经常忘记抛射角是相对于水平方向测量的，而不是相对于竖直方向。

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2. Independence of Horizontal and Vertical Motion | 水平与竖直运动的独立性

The key idea in projectile motion is that the horizontal and vertical motions are independent. Gravity acts only vertically, so it changes the vertical velocity but has no effect on the horizontal velocity. This allows us to analyse the two directions separately using different equations of motion.

抛体运动的关键思想是水平方向和竖直方向的运动相互独立。重力只作用于竖直方向，因此它改变竖直速度，但对水平速度没有影响。这使我们可以用不同的运动方程分别分析两个方向。

In the horizontal direction there is no acceleration if air resistance is ignored: aₓ = 0. In the vertical direction the acceleration is constant: a_y = -g, where g is the acceleration due to gravity, taken as 9.81 m s⁻² near the Earth's surface. The minus sign indicates that the acceleration acts downwards.

在水平方向上，如果忽略空气阻力，加速度为零：aₓ = 0。在竖直方向上，加速度是恒定的：a_y = -g，其中 g 是重力加速度，在地球表面附近取为 9.81 m s⁻²。负号表示加速度方向向下。

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3. Horizontal Motion: Constant Velocity | 水平运动：匀速运动

Because there is no horizontal acceleration, the horizontal velocity stays constant throughout the flight. The horizontal displacement is therefore found using the simple equation for constant velocity. This is true as long as air resistance is negligible.

由于没有水平加速度，水平速度在整个飞行过程中保持不变。因此水平位移可以用匀速运动的简单公式计算。只要空气阻力可以忽略，这个结论就成立。

x = uₓ t = u cos θ × t

In this equation x is the horizontal displacement, uₓ is the constant horizontal velocity, and t is the time of flight. Many exam questions require you to find the time from the vertical motion first, and then use that time to find the horizontal distance travelled.

在这个方程中，x 是水平位移，uₓ 是恒定的水平速度，t 是飞行时间。许多考试题目要求你先从竖直运动求出时间，然后再用该时间求水平位移。

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4. Vertical Motion: Uniform Acceleration | 竖直运动：匀加速运动

The vertical motion of a projectile is simply uniformly accelerated motion under gravity. You can use the standard SUVAT equations, but you must be careful with the sign convention. If you take upward as positive, the acceleration is a = -g = -9.81 m s⁻². If you take downward as positive, then a = +9.81 m s⁻².

抛体的竖直运动只是重力作用下的匀加速运动。你可以使用标准的 SUVAT 方程，但必须注意符号规则。如果取向上为正，加速度 a = -g = -9.81 m s⁻²。如果取向下为正，则 a = +9.81 m s⁻²。

The three most useful vertical equations are:

竖直方向最常用的三个方程是：

v_y = u_y - gt

y = u_y t - ½ gt²

v_y² = u_y² - 2gy

Here y is the vertical displacement, u_y is the initial vertical velocity, v_y is the final vertical velocity, and t is the time. Always decide which direction is positive before substituting values.

这里的 y 是竖直位移，u_y 是初始竖直速度，v_y 是末竖直速度，t 是时间。在代入数值之前，一定要先确定哪个方向为正。

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5. Time of Flight on Level Ground | 水平地面上的飞行时间

For a projectile launched from and landing on the same horizontal level, the total vertical displacement is zero at the end of the flight. We can set y = 0 in the vertical displacement equation and solve for t. This gives two solutions: the initial time t = 0 and the final time of flight.

对于从水平地面发射并落在同一水平面上的抛体，飞行结束时总竖直位移为零。我们可以在竖直位移方程中令 y = 0，然后解出 t。这样会得到两个解：初始时刻 t = 0 和最终的飞行时间。

0 = u_y T - ½ g T²

T = 2u_y / g = 2u sin θ / g

This equation shows that the time of flight depends only on the initial vertical velocity and the acceleration due to gravity. The horizontal component does not affect how long the projectile stays in the air on level ground.

这个方程表明飞行时间只取决于初始竖直速度和重力加速度。水平分量不影响抛体在水平地面上停留的时间。

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6. Maximum Height | 最大高度

The projectile reaches its maximum height when the vertical velocity becomes zero. At the highest point, the object is still moving horizontally, because the horizontal velocity never changes. We can use the equation v_y = u_y - gt with v_y = 0 to find the time to reach the peak.

当竖直速度变为零时，抛体达到最大高度。在最高点，物体仍然在水平运动，因为水平速度从不改变。我们可以使用 v_y = u_y - gt，令 v_y = 0，求出到达最高点的时间。

t_peak = u_y / g = u sin θ / g

H = u_y² / (2g) = u² sin² θ / (2g)

The maximum height depends on the square of the initial vertical velocity. A projectile launched straight upwards with θ = 90° reaches the greatest possible height for a given speed, because all of the initial kinetic energy goes into vertical motion.

最大高度取决于初始竖直速度的平方。以 θ = 90° 竖直向上发射的抛体，在给定速度下能达到最大高度，因为所有初始动能都用于竖直运动。

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7. Range of a Projectile | 抛体的射程

The horizontal range is the total horizontal distance travelled when the projectile returns to its original height. To find it, multiply the constant horizontal velocity by the time of flight. This gives a very useful formula for level ground.

水平射程是抛体回到原来高度时所经过的总水平距离。要求射程，只需用恒定的水平速度乘以飞行时间。这样可以得到一个在水平地面上非常有用的公式。

R = uₓ T = u cos θ × (2u sin θ / g)

R = u² sin 2θ / g

This result shows that the maximum range on level ground occurs when sin 2θ = 1, which means 2θ = 90°, so θ = 45°. At that angle, horizontal and vertical components are balanced to maximise the range for a fixed launch speed.

这个结果表明，在水平地面上最大射程出现在 sin 2θ = 1 时，即 2θ = 90°，所以 θ = 45°。在这个角度下，水平分量和竖直分量达到平衡，使固定发射速度下的射程最大。

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8. Equation of the Trajectory | 轨迹方程

By eliminating time from the horizontal and vertical displacement equations, we can obtain the equation of the trajectory. This equation shows that the path of an ideal projectile is a parabola. It is useful for determining whether a projectile will clear an obstacle or hit a target.

通过消去水平位移和竖直位移方程中的时间，我们可以得到轨迹方程。这个方程表明理想抛体的路径是一条抛物线。它对于判断抛体是否能越过障碍物或命中目标非常有用。

y = x tan θ - g x² / (2 u² cos² θ)

This equation is in the form y = ax - bx², which is a parabola. The first term represents the straight-line path if gravity were absent, and the second term represents the downward curvature caused by gravity.

这个方程的形式是 y = ax - bx²，是一条抛物线。第一项表示没有重力时的直线路径，第二项表示由重力引起的向下弯曲。

Quantity	Formula	Meaning
Time of flight	T = 2u sin θ / g	Total time in the air on level ground
Maximum height	H = u² sin² θ / (2g)	Highest vertical displacement reached
Range	R = u² sin 2θ / g	Horizontal distance travelled on level ground

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9. Projectiles Launched from a Height | 从一定高度抛出的抛体

If a projectile is launched from a cliff or a raised platform, the final vertical displacement is not zero. Instead, the projectile lands a vertical distance h below its starting point, so we set y = -h if upward is positive. The time of flight is then found by solving a quadratic equation.

如果抛体从悬崖或高台抛出，末竖直位移不为零。相反，抛体落在起点下方竖直距离 h 处，因此如果取向上为正，我们令 y = -h。此时飞行时间通过求解二次方程得到。

-h = u_y t - ½ g t²

This equation can be rearranged into standard quadratic form and solved with the quadratic formula. You should discard any negative time solution. The horizontal range is then found by multiplying the horizontal velocity by the positive time of flight.

这个方程可以改写为标准二次方程形式，并用求根公式求解。你应该舍去任何负的时间解。然后用水平速度乘以正飞行时间即可得到水平射程。

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10. Effects of Air Resistance | 空气阻力的影响

In the real world, air resistance is often significant, especially for fast-moving or light projectiles. Air resistance acts opposite to the direction of motion and increases with speed. It reduces both the maximum height and the range, and it makes the trajectory asymmetric rather than a perfect parabola.

在现实世界中，空气阻力往往不可忽略，特别是对于快速运动或较轻的抛体。空气阻力的方向与运动方向相反，并随速度增加而增大。它同时降低最大高度和射程，并使轨迹不再是对称的抛物线。

With air resistance, the horizontal velocity decreases over time, so the range formula R = u² sin 2θ / g no longer applies. The optimum launch angle for maximum

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