Question

In Figure, a small, solid, uniform ball is to be shot from point P so that it rolls smoothly along a horizontal path, up along a ramp, and onto a plateau. Then it leaves the plateau horizontally to land on a game board, at a horizontal distance d from the right edge of the plateau. The vertical heights are${\mathit{h}}_{\mathbf{1}}\mathbf{=}\mathbf{5}\mathbf{.00}\mathbf{\text{\hspace{0.17em}cm}}$and ${\mathit{h}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{.60}\mathbf{\text{\hspace{0.17em}cm}}$. With what speed must the ball be shot at point P for it to land at$\mathit{d}\mathbf{=}\mathbf{6}\mathbf{.00}\mathbf{\text{\hspace{0.17em}cm}}$?

Short answer

Initial speed of ball is$1.34\text{m}/\text{s}$

Step-by-step solution

Identification of given data

i) Friction is zero

ii) Height $h_1=5.0\text{cm}\&h_2=1.60\text{cm}$and

iii) Horizontal distance,$d=6.00\text{cm}$

To understand the concept

As the problem says there is no friction, potential and kinetic energy is conserved. Therefore, first find the change in kinetic energy of the ball during upward motion. In addition, by using initial height, define the time taken by the ball to reach that height. With the help of that time,find initial velocity at that point. In addition, by using all calculated parameters, find initial velocity of ball.

Formulae:

$K.E.=\frac{1}{2}M{v}^2$

$P.E.=M\times g\times h$

$s=v_{0}t+\frac{1}{2}g{t}^2$

Determining the time taken by the ball to travel at a heighth2  from the top to the bottom

At initial position, the energy is conserved.

$\text{Initial}PE=0,$

$\text{Initial}KE=\frac{1}{2}\times M\times v_0^2$

As the ball moves to thetop, kinetic energy decreases and potential energy increases.

So,

$P.E.=M\times g\times h$

And new kinetic energy

$K.E.=\frac{1}{2}\times M\times v_0^2-M\times g\times h_1$

New kinetic energy at top

$K.E.=\frac{1}{2}\times M\times v_0^2-M\times g\times h_1$

We have,$h_1=5.00\text{cm}$and$g=9.80{\text{m/s}}^{\text{2}}$

$K.E.=\frac{1}{2}\times M\times v_0^2-M\times 0.049{\text{\hspace{0.17em}m}}^{\text{2}}{\text{/s}}^{\text{2}}$

As the ball leaves the plateau, its horizontal velocity remains constant. However, its vertical velocity increases from zero at the rate of$9.8\text{m/s}$per second. Now, we can findthetime to travel $0.016\text{\hspace{0.17em}m}$distance.

We can first find the time taken by the ball to travel at a height$h_2$from the top to the bottom. We can usethe second kinematic equation

$\begin{array}{c}{h}_2=v_0\text{t}+\frac{1}{2}g{t}^2\\ 0.016\text{\hspace{0.17em}m}=0+\frac{1}{2}\times 9.8{\text{\hspace{0.17em}m/s}}^{\text{2}}\times t^2\end{array}$

$\begin{array}{c}t=\sqrt{\frac{0.016\text{\hspace{0.17em}m}}{4.9{\text{\hspace{0.17em}m/s}}^{\text{2}}}}\\ =0.058\text{sec}\end{array}$

Determining the horizontal velocity to move  0.06 m in 0.058 sec.0.058 sec.

Now we can determine the horizontal velocity to move$0.06\text{m}$intheabove time.

We apply second kinematic equation here again

$s=v_{0}t+\frac{1}{2}g{t}^2$

$0.06\text{\hspace{0.17em}m}=\left(v_0\right)0.058\text{\hspace{0.17em}s}+0$

$v_0=\frac{0.06\text{\hspace{0.17em}m}}{0.058\text{s}}$

$\Rightarrow v_0=1.305\text{m}/\text{s}$

Therefore, we can find kinetic energy at height $h_2$as,

$K.E.=\frac{1}{2}\times M\times {\left(1.305\text{\hspace{0.17em}m/s}\right)}^2$

Determining the speed with which ball must be shot at point P for it to land at  d=6.00cm

Since there is no friction, the K.E. ofthe ball at the top is equal to kinetic energy as it leaves the plateau,

$\begin{array}{c}\frac{1}{2}\times M\times v_0^2-M\times 0.049{\text{\hspace{0.17em}m}}^{\text{2}}{\text{/s}}^{\text{2}}=\frac{1}{2}\times M\times {\left(1.305\text{\hspace{0.17em}m/s}\right)}^2\\ \frac{1}{2}\times v_0^2-0.049{\text{\hspace{0.17em}m}}^{\text{2}}{\text{/s}}^{\text{2}}=\frac{1}{2}\times {\left(1.305\text{\hspace{0.17em}m/s}\right)}^2\\ v_0^2-0.098={\left(1.305\text{\hspace{0.17em}m/s}\right)}^2\\ v_0=1.34\text{m}/\text{s}\end{array}$