Q: If an object is projected upward with an initial velocity of $$20\ \mathrm{m/s}$$, how long will it take to return to the ground? (Assume $$g = 9.8\ \mathrm{m/s^2$}$$ and ignore air resistance.)

$$t = \frac{2v_0}{g} = \frac{2 \times 20}{9.8} \approx 4.08\ \mathrm{s}$$

Q: A ball is thrown straight up with an initial velocity of $$15\ \mathrm{m/s}. $$What is the maximum height it reaches? (Take $$g = 9.8\ \mathrm{m/s^2$})$$

$$h_{max} = \frac{(15)^2$}{2 \times 9.8} \approx 11.5\ \mathrm{m}$$

Question 1

An object is thrown vertically upward from the ground with an initial velocity $$v_0. $$Ignoring air resistance, which of the following equations correctly gives the time $$t_{max} to $$reach the maximum height?

Accepted Answer

$$t_{max} = \frac{v_0$}{g}$$

Question 2

A ball is dropped from rest from a height $$h$$ above the ground. What is the velocity of the ball just before it hits the ground? (Ignore air resistance.)

Accepted Answer

$$v = \sqrt{2gh}$$

Question 3

If an object is projected upward with an initial velocity of $$20\ \mathrm{m/s}$$, how long will it take to return to the ground? (Assume $$g = 9.8\ \mathrm{m/s^2$}$$ and ignore air resistance.)

Accepted Answer

$$t = \frac{2v_0}{g} = \frac{2 	imes 20}{9.8} \approx 4.08\ \mathrm{s}$$

Question 4

Which of the following best describes the velocity of an object at the maximum height of its upward trajectory (ignoring air resistance)?

Accepted Answer

The velocity is zero.

Question 5

A rock is thrown vertically upward from the edge of a cliff with an initial velocity $$v_0. If $$the acceleration due to gravity is $$g$$, which equation gives the maximum height $$h_{max}$$ above the starting point?

Accepted Answer

$$h_{max} = \frac{v_0$^2}{2g}$$

Question 6

If you choose the upward direction as positive and an object is thrown upward, what is the sign of the acceleration due to gravity in your equations?

Accepted Answer

Negative

Question 7

A ball is thrown straight up with an initial velocity of $$15\ \mathrm{m/s}. $$What is the maximum height it reaches? (Take $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$h_{max} = \frac{(15)^2$}{2 	imes 9.8} \approx 11.5\ \mathrm{m}$$

Question 8

Which kinematic equation would you use to find the displacement $$\Delta y of an $$object under constant acceleration, given initial velocity $$v_0$$, time $$t$$, and acceleration $$a$$?

Accepted Answer

$$\Delta y = v_0$ t + \frac{1}{2} a t^2$

Question 9

A student throws a ball upward from the ground and it returns to the same point. If the total flight time is $$T$$, what is the time to reach the maximum height?

Accepted Answer

$$\frac{T}{2}$$

Question 10

An object is projected vertically upward with initial velocity $$v_0. $$Which of the following statements is true about the magnitude of the velocity just before it returns to the starting point?

Accepted Answer

It is equal to $$v_0.$$

Question 11

A ball is thrown upward from the ground with an initial velocity of $$12\ \mathrm{m/s}. $$How long does it take to reach its maximum height? (Use $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$t = \frac{12}{9.8} \approx 1.22\ \mathrm{s}$$

Question 12

Which of the following best explains why the time to rise to maximum height equals the time to fall back to the starting point for an object thrown vertically upward (ignoring air resistance)?

Accepted Answer

The acceleration due to gravity is constant and acts equally during ascent and descent.

Question 13

A ball is thrown upward from a balcony $$20\ \mathrm{m}$$ above the ground with an initial velocity of $$10\ \mathrm{m/s}. $$How long does it take to reach the ground? (Use $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

Solve $$0 = 20 + 10t - 4.9t^2$; t$$ \approx 3.13\ \mathrm{s}$$

Question 14

Which method can be used to find the time at which a projectile reaches its maximum height using calculus?

Accepted Answer

Set the first derivative of the position function with respect to time equal to zero.

Question 15

A ball is thrown vertically upward with an initial velocity $$v_0$$ from the ground. Which of the following equations gives the position $$y at $$time $$t$$?

Accepted Answer

$$y = v_0$ t - \frac{1}{2} g t^2$

Question 16

If an object is thrown upward and you want to find the time when it returns to its original position, which equation should you solve for $$t$$?

Accepted Answer

$$0 = v_0$ t - \frac{1}{2} g t^2$

Question 17

A ball is thrown upward with an initial velocity $$v_0$$ from the ground. What is the velocity of the ball after time $$t$$?

Accepted Answer

$$v = v_0$ - g t$$

Question 18

A student analyzes the motion of a ball thrown upward and claims that the acceleration is zero at the top of the trajectory. Is this correct?

Accepted Answer

No, the acceleration is always $$-g ($$downward) throughout the motion.

Question 19

Which of the following is a correct application of the quadratic formula to solve for the time when a projectile returns to its original position?

Accepted Answer

$$t = \frac{2v_0}{g} ($$for $$y_0 = 0$)

Vertical Motion Under Constant Acceleration: Kinematics of Free Fall

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