اختبار اختيار من متعدد تفاعلي: L6-5 Projectile Motion
Projectile motion involves analyzing the trajectory of an object launched into the air under the influence of gravity. In this study, we explore various scenarios such as divers dropping from heights, balls propelled upward, and objects launched at specific angles. Key parameters include initial velocity, launch angle, and vertical acceleration. Using kinematic equations and calculus, we can determine the time of flight, horizontal range, and the velocity of an object at impact. This set of problems provides practical applications of vertical and horizontal motion equations in both metric and imperial units, emphasizing the integration of mathematical models with physical phenomena.
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اختبار شهادة تدريبي مؤقت للصف والمادة والفصل نفسه.
اختر إجابة واحدة لكل سؤال. عند الاختيار ستظهر النتيجة فورًا: الأخضر صحيح، والأحمر خطأ، وسيظهر تفسير الإجابة مباشرة إن كان متوفرًا. وبعد آخر سؤال ستظهر الدرجة النهائية تلقائيًا.
A diver drops from a height of 64 feet. Which of the following gives the initial conditions?
تفسير الإجابة
The diver starts at a height of 64 ft, so h(0) = 64. Since the diver 'drops' (starting from rest), the initial velocity is zero, so \(h'(0) = 0\).
السؤال 2
النقاط: 1
A diver drops from a height of 64 feet. What is the velocity at impact?
تفسير الإجابة
Using \(v^2 = v_0^2 + 2g\Delta h\) with v0 = 0, g = 32 ft/s2, and \(\Delta h = 64 \text{ ft}\): \(v = \sqrt{2 \cdot 32 \cdot 64} = \sqrt{4096} = 64 \text{ ft/s}\).
السؤال 3
النقاط: 1
Identify the initial condition y(0) and \(y'(0)\) for the vertical motion, if the object is thrown at a velocity of 6 m/s from a height of 30 m. (Take the origin to be the ground)
تفسير الإجابة
The initial height is 30 m above the ground, so y(0) = 30. Since it is thrown upward with a velocity of 6 m/s, the initial vertical velocity \(y'(0) = 6\).
السؤال 4
النقاط: 1
For an object moving in two dimensions shown in figure below. What is the initial value of the vertical component of the velocity?
تفسير الإجابة
The vertical component of velocity is calculated using the sine of the launch angle: \(v_{0y} = v_0 \sin(\theta) = 110 \sin(60^\circ) = 110 \cdot \frac{\sqrt{3}}{2} = 55\sqrt{3}\).
السؤال 5
النقاط: 1
For an object moving in two dimensions shown in figure below. What is the vertical acceleration?
تفسير الإجابة
Since the velocity units in the problem are meters per second (m/s), the standard vertical acceleration due to gravity is -9.8 m/s2.
السؤال 6
النقاط: 1
If an object dropped from a height of H feet, at what time t it will hit the ground?
تفسير الإجابة
Using the equation \(h = \frac{1}{2}gt^2\) with g = 32 ft/s2: H = 16t2. Solving for t, we get t2 = H/16, so \(t = \sqrt{H}/4\).
السؤال 7
النقاط: 1
An object is launched at angle \(\theta = \frac{\pi}{3}\) radians from the horizontal with an initial speed of 98 m/s. Determine the time of flight.
An object is launched at angle \(\theta = \frac{\pi}{3}\) radians from the horizontal with an initial speed of 98 m/s. Which of the following give the horizontal range?
تفسير الإجابة
The horizontal position is \(x(t) = v_0 \cos(\theta)t\). Here \(x(t) = 98 \cos(60^\circ)t = 98 \cdot 0.5 \cdot t = 49t\).
السؤال 9
النقاط: 1
A diver drops from 30 ft above the water. Which of the following give the height of the diver at time t?
تفسير الإجابة
Using the equation \(h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\) with g = 32 ft/s2, v0 = 0, and h0 = 30, we get h(t) = -16t2 + 30.
السؤال 10
النقاط: 1
A ball is propelled straight upward from the ground with initial velocity 64 ft/s. Ignoring air resistance, determine the amount of time the ball spends in the air.
تفسير الإجابة
Total time in air \(T = \frac{2v_0}{g} = \frac{2 \cdot 64}{32} = \frac{128}{32} = 4 \text{ s}\).
السؤال 11
النقاط: 1
A certain not-so-wily coyote discovers that he just stepped off the edge of a cliff. Four seconds later, he hits the ground in a puff of dust. How high in meters was the cliff?
تفسير الإجابة
Using \(h = \frac{1}{2}gt^2\) in meters (g = 9.8 m/s2): h = 0.5 × 9.8 × 42 = 4.9 × 16 = 78.4 m.
السؤال 12
النقاط: 1
The coyote's next scheme involves launching himself into the air with an Acme catapult. If the coyote is propelled vertically from the ground with initial velocity 19.6 m/s, find an equation for the height of the coyote at any time t.
تفسير الإجابة
Using \(h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\) with g = 9.8 m/s2, v0 = 19.6, and h0 = 0: h(t) = -4.9t2 + 19.6t.
السؤال 13
النقاط: 1
The coyote's next scheme involves launching himself into the air with an Acme catapult. If the coyote is propelled vertically from the ground with initial velocity 19.6 m/s, find his maximum height.
تفسير الإجابة
Max height occurs when v = 0, which is at t = v0/g = 19.6/9.8 = 2 s. Substituting into the height equation: h(2) = -4.9(22) + 19.6(2) = -19.6 + 39.2 = 19.6 m.
السؤال 14
النقاط: 1
A baseball pitcher releases the ball horizontally from a height of 6 ft with an initial speed of 130 ft/s. Find the height of the ball when it reaches home plate 60 feet away.
تفسير الإجابة
Horizontal time t = x/v_x = 60/130 = 6/13 s. Vertical position y(t) = -16t2 + 6. So, \(y(6/13) = -16(6/13)^2 + 6 = -16(36/169) + 6 = -576/169 + 6 \approx -3.408 + 6 = 2.592 \text{ ft}\).
السؤال 15
النقاط: 1
A plane at an altitude of 256 feet want to drop supplies to a specific location on the ground. If the plane has a horizontal velocity of 100 ft/s, how far away from the target should the plane release the supplies in order to hit the target location?
تفسير الإجابة
First, find the time to hit the ground: \(0 = -16t^2 + 256 \rightarrow 16t^2 = 256 \rightarrow t^2 = 16 \rightarrow t = 4 \text{ s}\). The horizontal distance D = v_x × t = 100 × 4 = 400 ft.
السؤال 16
النقاط: 1
An object is launched from the ground at an angle \(\theta = 20^\circ\) with an initial speed of 48 m/s. Find the time of the flight? (ignore the air resistance)
تفسير الإجابة
If we use g = 32 (implied by the options despite the m/s unit), \(T = 2v_0 \sin(\theta)/g = 2(48)\sin(20^\circ)/32 = 96 \cdot 0.342 / 32 = 1.026 \text{ s}\).
السؤال 17
النقاط: 1
One of the authors has a vertical 'jump' of 20 in. What is the initial velocity required to jump this high?
تفسير الإجابة
Using v2 = v02 - 2gh, at max height v = 0. h = 20/12 = 5/3 ft. \(v_0 = \sqrt{2 \cdot 32 \cdot 5/3} = \sqrt{320/3} \approx 10.33 \text{ ft/s}\).
السؤال 18
النقاط: 1
A diver drops from 120 feet above the water (about the height of divers at the Acapulco Cliff Diving competition). What is the diver's velocity at impact?
تفسير الإجابة
Using \(v^2 = v_0^2 + 2g\Delta y\) with v0 = 0, g = 32, and \(\Delta y = -120\): \(v = -\sqrt{2 \cdot 32 \cdot 120} = -\sqrt{7680} = -\sqrt{256 \cdot 30} = -16\sqrt{30} \text{ ft/s}\). The negative sign indicates downward direction.
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