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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 7.3 Practice - Constant Acceleration

Exercises Practice Activities

A*R*C*S 7.3.1. Pitching Speed.

A baseball player wants to determine his pitching speed. You have him stand on a ledge and throw the ball horizontally from an elevation \(5 \mathrm{~m}\) above the ground. The ball hits the ground \(30 \mathrm{~m}\) away. What is his pitching speed?
Solution.
1. Analyze and Represent
  1. \begin{equation*} y_i = 5 \mathrm{~m} \end{equation*}
    \begin{equation*} y_f = 0 \end{equation*}
    \begin{equation*} x_i = 0 \end{equation*}
    \begin{equation*} x_f = 30 \mathrm{~m} \end{equation*}
    \begin{equation*} a_y = -g = -10 \mathrm{~m/s^2} \end{equation*}
    \begin{equation*} v_{ix} = ? \end{equation*}
  2. Particle model: ignore rotation of the ball. Near Earth with gravity as the only force, leading to an acceleration of \(-g\hat{y}\text{.}\)
  3. A simple physical picture will probably do here.
    Figure 7.3.1. A baseball’s path.
2. Calculate
  1. \begin{equation*} x_f = x_{i} + v_{ix}t \end{equation*}
    \begin{equation*} y_f = y_{i} + v_{iy}t + \frac{1}{2}a_y t^2 \end{equation*}
  2. \begin{equation*} 0 = y_{i} - \frac{1}{2}g t^2 \end{equation*}
    \begin{equation*} t = \sqrt{\frac{2y_i}{g}} \end{equation*}
    \begin{equation*} x_f = v_{ix}t = v_{ix}\sqrt{\frac{2y_i}{g}} \end{equation*}
    \begin{equation*} v_{ix} = x_f\sqrt{\frac{g}{2y_i}} \end{equation*}
  3. \begin{equation*} v_{ix} = 30 \mathrm{~m/s} \end{equation*}
3. Sensemake
  1. \begin{equation*} \mathrm{m/s} = \mathrm{m}\sqrt{\frac{\mathrm{m/s^2}}{\mathrm{m}}} = \mathrm{m/s} \end{equation*}
  2. \(30 \mathrm{~m/s}\) is about \(65 \mathrm{~mph}\text{.}\) Professional baseball pitchers regularly reach around \(100 \mathrm{~mph}\text{,}\) so while my friend is doing okay, he might need a little more practice before going pro.
  3. \(v_{ix}\) depends on three quantities, two of which are of interest: \(y_i\) and \(x_f\text{.}\) For \(x_f\text{,}\) if you are able to throw the ball a greater horizontal distance without it falling farther, you must be throwing it faster. Sure enough, \(x_f\) is in the numerator of our expression, so this agrees with our physical reasoning.
    On the other hand, a larger value of \(y_i\) means that the baseball will fall a greater distance without actually traveling any farther horizontally. This can only be true if the ball is thrown with a smaller initial speed. The equation has \(y_i\) in the denominator, so once again this agrees!

A*R*C*S 7.3.2. Stopping a Car.

You are driving a car on a residential street at a speed of about \(v = 10 \mathrm{~m/s}\) when a ball bounces in front of you and you slam on the brakes, leading to an acceleration with a magnitude of about \(a = 12.5 \mathrm{~m/s^2}\text{.}\) How much time does it take for you to stop? How much distance have you traveled in this time?

Explanation 7.3.3. Ramp Race.

You release a ball from rest at the top of a ramp and measure the amount of time for the ball to reach the base of the ramp. Your friend performs the same experiment (with the same ball) with their ramp, which is the same length as your ramp. Your friend determines that it took more time for the ball to reach the bottom of their ramp than it did for the ball to reach the bottom of your ramp. Is the magnitude of the average acceleration of the ball for your friend’s ramp greater than, less than, or equal to the magnitude of the average acceleration of the ball for your ramp?

A*R*C*S 7.3.4. Throwing at a Cliff.

A ball is thrown toward a cliff of height \(h\) with a speed of \(27 \mathrm{~m/s}\) and an angle of \(60^o\) above the horizontal. It lands on the edge of the cliff \(3.2 \mathrm{~s}\) later. How high is the cliff?
Tip.
When making sense of your answer, you should often look to add something to your knowledge or confidence in your answer. Reiterating the process you used to solve the problem does not count!

Explanation 7.3.5. Throwing It Back.

In the previous activity, you considered a ball thrown up to the top of a cliff. Suppose you wanted to throw the ball back from the top of the cliff so that it took the same amount of time to travel to the place where it was originally thrown from. Is it possible to do this? If so, how? If not, why not? (Note: you do not need to find any numbers: let your reasoning do the talking!)
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