Question

In shot putting, many athletes elect to launch the shot at an angle that is smaller than the theoretical one (about $\mathbf{42}\mathbf{°}$) at which the distance of a projected ball at the same speed and height is greatest. One reason has to do with the speed the athlete can give the shot during the acceleration phase of the throw. Assume that a 7.260 kgshot is accelerated along a straight path of length 1.650 mby a constant applied force of magnitude 380.0 N, starting with an initial speed of 2.500 m/s(due to the athlete’s preliminary motion). (a)What is the shot’s speed at the end of the acceleration phase if the angle between the path and the horizontal is 30.00° and (b)What is the shot’s speed at the end of the acceleration phase if the angle between the path and the horizontal is 42.00°? (Hint:Treat the motion as though it were along a ramp at the given angle.) (c) By what percent is the launch speed decreased if the athlete increases the angle from 30.00° to42.00°?

Short answer

1. The speed of the shot at the end of the acceleration phase if the angle between the path and the horizontal is 30° is 12.76 m/s
2. The speed of the shot at the end of the acceleration phase if the angle between the path and the horizontal is 42° is 12.54 m/s
3. The percentage decrease in the launch speed if the athlete increases the angle 30° from 42° is 1.724%.

Step-by-step solution

Given information

It is given that,

The mass of the shot is M=7.26 kg

The distance travelled by the shot x=1.65 m

The force applied on the shot F=380 N.

The initial velocity of the shot$v_0=2.5m/s$

Determining the concept

The problem is based on Newton’s second law of motion which states that the rate of change of momentum of a body is equal in both magnitude and direction of the force acting on it. Thus, using the free body diagram and also kinematic equation, the final velocity of the shot at different anglescan be found.

Formulae:

Newton’s second law is,

$F_{net}=\sum _{}Ma$ (i)

where, $F_{net}$ is the net force, Mis mass and a is an acceleration.

Newton’s third kinematic equation is,

$v_f^2=v_0^2+2a\left(∆x\right)$ (ii)

where, $v_f$is the final velocity, $v_0$is the initial velocity, a is an acceleration and$∆x$ is the displacement.

(a) Determining the speed of the shot at the end of the acceleration phase if the angle between the path and the horizontal is 30°

FBD for the system is,

From FBD, write for the net force acting in x-direction as,

$$\begin{gathered} F_{netx}=F-Mg\sin \theta \\ {F}_{netx}=380-(7.260)(9.8)\sin 30° \\ {F}_{netx}=344.4N \end{gathered}$$

Using equation (i),

$$\begin{gathered} a_x=\frac{F_{netx}}{M} \\ {a}_x=\frac{344.4}{7.26} \\ {a}_x=47.44m/s^2 \end{gathered}$$

Now, with equation (ii), final velocity will be,

$v_f=\sqrt{{v_0}^2+2a\left(∆x\right)}$

So,the speed of the shot at the end of the acceleration phase if the angle between the path and the horizontal 30° is

$$\begin{gathered} v_f=\sqrt{2.5^2+2\left(47.44\right)\left(1.65\right)} \\ {v}_f=12.76m/s \end{gathered}$$

So, the speed of the shot at the end of the acceleration phase if the angle between the path and the horizontal 30° is $v_f=12.76m/s$

(b) Determining the speed of the shot at the end of the acceleration phase if the angle between the path and the horizontal is 42°

According to Newton’s second law,

At $\theta ={42}^o$

$$\begin{gathered} F_{netx}=F-Mg\sin \theta \\ {F}_{netx}=380-\left(7.260\right)\left(9.8\right)\sin 42° \\ {F}_{netx}=332.4N \end{gathered}$$

But,

$a_x=\frac{F_{netx}}{M}$

So,

$$\begin{gathered} a_x=\frac{332.4}{7.26} \\ {a}_x=45.78m/s^2 \end{gathered}$$

So, the speed of the shot at the end of the acceleration phase if the angle between the path and the horizontal is 42° is

$$\begin{gathered} v_f=\sqrt{2.5^2+2\left(45.78\right)\left(1.65\right)} \\ {v}_f=12.54m/s \end{gathered}$$

So, the speed of the shot at the end of the acceleration phase if the angle between the path and the horizontal is 42° is 12.54 m/s.

(c) Determining the percentage decrease in the launch speed if the athlete increases the angle from 30° to 42° is 1.69%

The percentage decrease in the launch speed if the athlete increases the angle from30°to 42° is,

$\frac{12.76-12.54}{12.76}\times 100=1.724\%$

The percentage decrease in the launch speed if the athlete increases the angle from30°to 42° is 1.724%.