1. A ball of mass m rolls off of a y-meter-high lab table and hits the floor a distance x from the base of the table.

Question

a. Show that the ball takes [latex]\sqrt{2 y / g}[/latex] seconds to hit the floor.

b. Show that the ball leaves the table at a speed [latex]x / \sqrt{2 y / g}[/latex] meters per second.

c. The ball has a mass of 0.010 kg, the height of the table is 1.25 m, and the ball hits the floor 3.0 m from the base of the table. Using [latex]g = 10 m/s^2[/latex] , show that the speed of the ball leaving the table is 6.0 m/s.

2. A horizontally moving tennis ball barely clears the net, a distance y above the surface of the court. To land within the tennis court the ball must not be moving too fast.

a. To remain within the court’s border a horizontal distance d from the bottom of the net, show that the ball’s maximum speed over the net is [latex]v=\frac{d}{\sqrt{\frac{2 y}{g}}}[/latex]

b. Suppose the height of the net is 1.00 m, and the court’s border is 12.0 m from the bottom of the net. Use [latex]g = 10 m/s^2[/latex] and show that the maximum speed of the horizontally moving ball clearing the net is about 27 m/s (about 60 mi/h).

c. Does the mass of the ball make a difference? Defend your answer.

Accepted Answer

1. a. We want the time of the ball in the air. First, some physics. The time t it takes for the ball to hit the floor would be the same as if it were dropped from rest a vertical distance y. We say from rest because the initial vertical component of velocity is zero.

[latex] ext { From } y=\frac{1}{2} g t^{2} \Rightarrow t^{2}=\frac{2 y}{g} \Rightarrow t=\sqrt{\frac{2 y}{g}}[/latex]

b. The horizontal speed of the ball as it leaves the table, using the time found in (a), is

[latex]v=\frac{d}{t}=\frac{x}{t}=\frac{x}{\sqrt{\frac{2 y}{g}}}[/latex].

c. [latex]v=\frac{x}{\sqrt{\frac{2 y}{g}}}=\frac{3.0 m }{\sqrt{\frac{2(1.25 m )}{10 \frac{ m }{ s ^{2}}}}}= 6 . 0 \frac{ m }{ s } .[/latex]

Notice how the terms of the equations guide the solution. Notice also that the mass of the ball, not showing up in the equations, is extraneous information (as would be the color of the ball).

2. a. As with Problem 1, the physics concept here involves projectile motion in the absence of air resistance, where horizontal and vertical components of velocity are independent. We’re asked for horizontal speed, so we write,

[latex]v_{ x }=\frac{d}{t}[/latex]

where d is horizontal distance traveled in time t. As with Problem 1, the time t of the ball in flight will be the same as if we had just dropped it from rest a vertical distance y from the top of the net. As the ball clears the net, its highest point in its path, its vertical component of velocity is zero.

[latex] ext { From } y=\frac{1}{2} g t^{2} \Rightarrow t^{2}=\frac{2 y}{g} \Rightarrow t=\sqrt{\frac{2 y}{g}}[/latex].

[latex] ext { So } v=\frac{d}{t}=\frac{d}{\sqrt{\frac{2 y}{g}}} ext {. }[/latex]

Can you see that solving in terms of symbols better shows that these two problems are one in the same? All the physics occurs in parts (a) and (b) in Problem 1. These steps are combined in part (a) of Problem 2.

b. [latex]v=\frac{d}{\sqrt{\frac{2 y}{g}}}=\frac{12.0 m }{\sqrt{\frac{2(1.00 m )}{10 \frac{ m }{ s ^{2}}}}}=26.8 \frac{ m }{ s } \approx 27 \frac{ m }{ s }[/latex].

c. We can see that the mass of the ball (in both problems) doesn’t show up in the equations for motion, which tells us that mass is irrelevant. Recall from that mass has no effect on a freely falling object—and the tennis ball is a freely falling object (as is every projectile when air resistance can be neglected).

Question 7.PS.1: 1. A ball of mass m rolls off of a y-meter-high lab table an...

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