Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 3, Problem 102P

(a)

To determine

The value of angle formed by initial velocity vector with horizontal.

(a)

Expert Solution
Check Mark

Answer to Problem 102P

  θo=86.38°

Explanation of Solution

Given data:

Final height of ball just above the plate =0.7 m

Range =18.4 m

Initial speed of ball =37.5 m/s

Formula used:

Vertical position of ball in terms of time t .

  y=yo+(vosinθo)t12gt2

Calculation:

The horizontal distance covered is given as

  x=(vocosθo)t                      ...(1)

The vertical position can be written as

  y=yo+(vosinθo)t12gt2

From equation (1), put t=xvocosθo

  y=yo+(vosinθo)t12gt2y=yo+(vosinθo)xvocosθo12g( x v o cos θ o )2y=yo(tanθo)xg2vo2 cos2θox2                      ...(2)

Now, substitute the values of respective parameters in equation

  y=yo(tanθo)xg2vo2 cos2θox20.7=2.0(tanθo)×18.49.812× ( 37.5 )2× cos2θo(18.4)218.4×tanθo+1.181 cos2θo=1.318.4×tanθo+1.181 cos2θo=1.318.4× sec2θo1+1.181 cos2θo=1.318.4× 1 cos 2 θ o1+1.181 cos2θo=1.318.4× 1 cos 2 θ o1=1.31.181 cos2θo

By squaring both sides,

  (18.4× 1 cos 2 θ o 1)2=(1.3 1.181 cos 2 θ o )2338.56 cos 2θo338.56=1.69+1.394 cos4θo3.07 cos2θo1.394 cos4θo341.63 cos2θo+340.25=0

After solving the above equation, the value of θo

  1.394 cos4θo341.63 cos2θo+340.25=01 cos2θo=( 341.63)± ( 341.63 ) 2 4×1.394×340.252×1.3941 cos2θo=341.63± 116711.05691897.2342.71 cos2θo=341.63±338.842.7

By taking the positive sign

  1 cos2θo=341.63+338.842.71 cos2θo=252.025cos2θo=1252.025cos2θo=3.967×103cosθo=3.967× 10 3cosθo=0.06299θo=cos1(0.06299)θo=86.38°

Conclusion:

The velocity and the horizon made an angle is θo=86.38°

(b)

To determine

The value of speed with which ball crosses plate.

(b)

Expert Solution
Check Mark

Answer to Problem 102P

  v=38.88 m/s

Explanation of Solution

Given data:

Final height of ball just above the plate =0.7 m

Range =18.4 m

Initial speed of ball =37.5 m/s

Formula used:

Speed is given by

  v=vx2+vy2

Calculation:

The horizontal component of velocity is vx

  vx=dxdt=ddt(vocosθo)t=vocosθo

The vertical component of velocity is vy

  vy=dydt=ddt(yo+vosinθo×t12gt2)=vosinθogt

Substitute these values into speed formula

  v= ( v o cos θ o )2+ ( v o sin θ o gt )2v= ( v o cos θ o )2+ ( v o sin θ o g x v o cos θ o )2v= ( 37.5×cos86.38° )2+ ( 37.5×sin86.38°9.81× 18.4 37.5×cos86.38° )2v=5.606+1506.2566v=38.88 m/s

Conclusion:

The velocity of the ball over the plate is 38.88 m/s .

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Physics for Scientists and Engineers

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