Where: [latex]u[/latex] = initial velocity, [latex]\theta[/latex] = angle of projection, [latex]g[/latex] = acceleration due to gravity (9.8 m/s²)
A projectile is launched at an angle of 30° with an initial velocity of 20 m/s. Find the maximum height, range, and time of flight.
Given: [latex]u = 20[/latex] m/s, [latex]\theta = 30°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(20)^2 \sin^2 30°}{2 \times 9.8}[/latex]
[latex]H = \frac{400 \times (0.5)^2}{19.6} = \frac{400 \times 0.25}{19.6} = \frac{100}{19.6} = 5.10[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(20)^2 \sin 60°}{9.8}[/latex]
[latex]R = \frac{400 \times 0.866}{9.8} = \frac{346.4}{9.8} = 35.35[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 20 \times \sin 30°}{9.8}[/latex]
[latex]T = \frac{40 \times 0.5}{9.8} = \frac{20}{9.8} = 2.04[/latex] s
A ball is thrown at 45° with initial velocity 25 m/s. Calculate maximum height, range, and time of flight.
Given: [latex]u = 25[/latex] m/s, [latex]\theta = 45°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(25)^2 \sin^2 45°}{2 \times 9.8}[/latex]
[latex]H = \frac{625 \times (0.707)^2}{19.6} = \frac{625 \times 0.5}{19.6} = \frac{312.5}{19.6} = 15.95[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(25)^2 \sin 90°}{9.8}[/latex]
[latex]R = \frac{625 \times 1}{9.8} = \frac{625}{9.8} = 63.78[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 25 \times \sin 45°}{9.8}[/latex]
[latex]T = \frac{50 \times 0.707}{9.8} = \frac{35.35}{9.8} = 3.61[/latex] s
A projectile is fired at 60° with speed 30 m/s. Find maximum height, range, and flight time.
Given: [latex]u = 30[/latex] m/s, [latex]\theta = 60°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(30)^2 \sin^2 60°}{2 \times 9.8}[/latex]
[latex]H = \frac{900 \times (0.866)^2}{19.6} = \frac{900 \times 0.75}{19.6} = \frac{675}{19.6} = 34.44[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(30)^2 \sin 120°}{9.8}[/latex]
[latex]R = \frac{900 \times 0.866}{9.8} = \frac{779.4}{9.8} = 79.53[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 30 \times \sin 60°}{9.8}[/latex]
[latex]T = \frac{60 \times 0.866}{9.8} = \frac{51.96}{9.8} = 5.30[/latex] s
A stone is thrown at 37° with initial velocity 15 m/s. Calculate the required parameters.
Given: [latex]u = 15[/latex] m/s, [latex]\theta = 37°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(15)^2 \sin^2 37°}{2 \times 9.8}[/latex]
[latex]H = \frac{225 \times (0.602)^2}{19.6} = \frac{225 \times 0.362}{19.6} = \frac{81.45}{19.6} = 4.16[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(15)^2 \sin 74°}{9.8}[/latex]
[latex]R = \frac{225 \times 0.961}{9.8} = \frac{216.23}{9.8} = 22.07[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 15 \times \sin 37°}{9.8}[/latex]
[latex]T = \frac{30 \times 0.602}{9.8} = \frac{18.06}{9.8} = 1.84[/latex] s
A projectile is launched at 53° with velocity 40 m/s. Find all parameters.
Given: [latex]u = 40[/latex] m/s, [latex]\theta = 53°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(40)^2 \sin^2 53°}{2 \times 9.8}[/latex]
[latex]H = \frac{1600 \times (0.799)^2}{19.6} = \frac{1600 \times 0.638}{19.6} = \frac{1020.8}{19.6} = 52.08[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(40)^2 \sin 106°}{9.8}[/latex]
[latex]R = \frac{1600 \times 0.961}{9.8} = \frac{1537.6}{9.8} = 156.90[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 40 \times \sin 53°}{9.8}[/latex]
[latex]T = \frac{80 \times 0.799}{9.8} = \frac{63.92}{9.8} = 6.52[/latex] s
A ball is projected at 20° with speed 18 m/s. Calculate maximum height, range, and time of flight.
Given: [latex]u = 18[/latex] m/s, [latex]\theta = 20°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(18)^2 \sin^2 20°}{2 \times 9.8}[/latex]
[latex]H = \frac{324 \times (0.342)^2}{19.6} = \frac{324 \times 0.117}{19.6} = \frac{37.91}{19.6} = 1.93[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(18)^2 \sin 40°}{9.8}[/latex]
[latex]R = \frac{324 \times 0.643}{9.8} = \frac{208.33}{9.8} = 21.26[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 18 \times \sin 20°}{9.8}[/latex]
[latex]T = \frac{36 \times 0.342}{9.8} = \frac{12.31}{9.8} = 1.26[/latex] s
A projectile is fired at 70° with initial velocity 35 m/s. Find the parameters.
Given: [latex]u = 35[/latex] m/s, [latex]\theta = 70°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(35)^2 \sin^2 70°}{2 \times 9.8}[/latex]
[latex]H = \frac{1225 \times (0.940)^2}{19.6} = \frac{1225 \times 0.884}{19.6} = \frac{1082.9}{19.6} = 55.25[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(35)^2 \sin 140°}{9.8}[/latex]
[latex]R = \frac{1225 \times 0.643}{9.8} = \frac{787.68}{9.8} = 80.37[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 35 \times \sin 70°}{9.8}[/latex]
[latex]T = \frac{70 \times 0.940}{9.8} = \frac{65.8}{9.8} = 6.71[/latex] s
A stone is thrown at 15° with velocity 12 m/s. Calculate all required values.
Given: [latex]u = 12[/latex] m/s, [latex]\theta = 15°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(12)^2 \sin^2 15°}{2 \times 9.8}[/latex]
[latex]H = \frac{144 \times (0.259)^2}{19.6} = \frac{144 \times 0.067}{19.6} = \frac{9.65}{19.6} = 0.49[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(12)^2 \sin 30°}{9.8}[/latex]
[latex]R = \frac{144 \times 0.5}{9.8} = \frac{72}{9.8} = 7.35[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 12 \times \sin 15°}{9.8}[/latex]
[latex]T = \frac{24 \times 0.259}{9.8} = \frac{6.22}{9.8} = 0.63[/latex] s
A projectile is launched at 75° with speed 28 m/s. Find maximum height, range, and flight time.
Given: [latex]u = 28[/latex] m/s, [latex]\theta = 75°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(28)^2 \sin^2 75°}{2 \times 9.8}[/latex]
[latex]H = \frac{784 \times (0.966)^2}{19.6} = \frac{784 \times 0.933}{19.6} = \frac{731.47}{19.6} = 37.32[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(28)^2 \sin 150°}{9.8}[/latex]
[latex]R = \frac{784 \times 0.5}{9.8} = \frac{392}{9.8} = 40.00[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 28 \times \sin 75°}{9.8}[/latex]
[latex]T = \frac{56 \times 0.966}{9.8} = \frac{54.10}{9.8} = 5.52[/latex] s
A ball is projected at 40° with initial velocity 22 m/s. Calculate the parameters.
Given: [latex]u = 22[/latex] m/s, [latex]\theta = 40°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(22)^2 \sin^2 40°}{2 \times 9.8}[/latex]
[latex]H = \frac{484 \times (0.643)^2}{19.6} = \frac{484 \times 0.413}{19.6} = \frac{199.89}{19.6} = 10.20[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(22)^2 \sin 80°}{9.8}[/latex]
[latex]R = \frac{484 \times 0.985}{9.8} = \frac{476.74}{9.8} = 48.65[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 22 \times \sin 40°}{9.8}[/latex]
[latex]T = \frac{44 \times 0.643}{9.8} = \frac{28.29}{9.8} = 2.89[/latex] s
A projectile is fired at 25° with velocity 50 m/s. Find all parameters.
Given: [latex]u = 50[/latex] m/s, [latex]\theta = 25°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(50)^2 \sin^2 25°}{2 \times 9.8}[/latex]
[latex]H = \frac{2500 \times (0.423)^2}{19.6} = \frac{2500 \times 0.179}{19.6} = \frac{447.5}{19.6} = 22.83[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(50)^2 \sin 50°}{9.8}[/latex]
[latex]R = \frac{2500 \times 0.766}{9.8} = \frac{1915}{9.8} = 195.41[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 50 \times \sin 25°}{9.8}[/latex]
[latex]T = \frac{100 \times 0.423}{9.8} = \frac{42.3}{9.8} = 4.32[/latex] s
A stone is thrown at 55° with speed 16 m/s. Calculate maximum height, range, and time of flight.
Given: [latex]u = 16[/latex] m/s, [latex]\theta = 55°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(16)^2 \sin^2 55°}{2 \times 9.8}[/latex]
[latex]H = \frac{256 \times (0.819)^2}{19.6} = \frac{256 \times 0.671}{19.6} = \frac{171.78}{19.6} = 8.76[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(16)^2 \sin 110°}{9.8}[/latex]
[latex]R = \frac{256 \times 0.940}{9.8} = \frac{240.64}{9.8} = 24.55[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 16 \times \sin 55°}{9.8}[/latex]
[latex]T = \frac{32 \times 0.819}{9.8} = \frac{26.21}{9.8} = 2.67[/latex] s
A projectile is launched at 35° with initial velocity 42 m/s. Find the parameters.
Given: [latex]u = 42[/latex] m/s, [latex]\theta = 35°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(42)^2 \sin^2 35°}{2 \times 9.8}[/latex]
[latex]H = \frac{1764 \times (0.574)^2}{19.6} = \frac{1764 \times 0.329}{19.6} = \frac{580.36}{19.6} = 29.61[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(42)^2 \sin 70°}{9.8}[/latex]
[latex]R = \frac{1764 \times 0.940}{9.8} = \frac{1658.16}{9.8} = 169.20[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 42 \times \sin 35°}{9.8}[/latex]
[latex]T = \frac{84 \times 0.574}{9.8} = \frac{48.22}{9.8} = 4.92[/latex] s
A ball is projected at 80° with velocity 24 m/s. Calculate all required values.
Given: [latex]u = 24[/latex] m/s, [latex]\theta = 80°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(24)^2 \sin^2 80°}{2 \times 9.8}[/latex]
[latex]H = \frac{576 \times (0.985)^2}{19.6} = \frac{576 \times 0.970}{19.6} = \frac{558.72}{19.6} = 28.51[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(24)^2 \sin 160°}{9.8}[/latex]
[latex]R = \frac{576 \times 0.342}{9.8} = \frac{197.00}{9.8} = 20.10[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 24 \times \sin 80°}{9.8}[/latex]
[latex]T = \frac{48 \times 0.985}{9.8} = \frac{47.28}{9.8} = 4.82[/latex] s
A projectile is fired at 10° with speed 60 m/s. Find maximum height, range, and flight time.
Given: [latex]u = 60[/latex] m/s, [latex]\theta = 10°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(60)^2 \sin^2 10°}{2 \times 9.8}[/latex]
[latex]H = \frac{3600 \times (0.174)^2}{19.6} = \frac{3600 \times 0.030}{19.6} = \frac{108}{19.6} = 5.51[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(60)^2 \sin 20°}{9.8}[/latex]
[latex]R = \frac{3600 \times 0.342}{9.8} = \frac{1231.2}{9.8} = 125.63[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 60 \times \sin 10°}{9.8}[/latex]
[latex]T = \frac{120 \times 0.174}{9.8} = \frac{20.88}{9.8} = 2.13[/latex] s
A stone is thrown at 65° with initial velocity 32 m/s. Calculate the parameters.
Given: [latex]u = 32[/latex] m/s, [latex]\theta = 65°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(32)^2 \sin^2 65°}{2 \times 9.8}[/latex]
[latex]H = \frac{1024 \times (0.906)^2}{19.6} = \frac{1024 \times 0.821}{19.6} = \frac{840.70}{19.6} = 42.89[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(32)^2 \sin 130°}{9.8}[/latex]
[latex]R = \frac{1024 \times 0.766}{9.8} = \frac{784.38}{9.8} = 80.04[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 32 \times \sin 65°}{9.8}[/latex]
[latex]T = \frac{64 \times 0.906}{9.8} = \frac{57.98}{9.8} = 5.92[/latex] s
A projectile is launched at 50° with velocity 26 m/s. Find all parameters.
Given: [latex]u = 26[/latex] m/s, [latex]\theta = 50°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(26)^2 \sin^2 50°}{2 \times 9.8}[/latex]
[latex]H = \frac{676 \times (0.766)^2}{19.6} = \frac{676 \times 0.587}{19.6} = \frac{396.81}{19.6} = 20.25[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(26)^2 \sin 100°}{9.8}[/latex]
[latex]R = \frac{676 \times 0.985}{9.8} = \frac{665.86}{9.8} = 67.95[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 26 \times \sin 50°}{9.8}[/latex]
[latex]T = \frac{52 \times 0.766}{9.8} = \frac{39.83}{9.8} = 4.07[/latex] s
A ball is projected at 85° with speed 14 m/s. Calculate maximum height, range, and time of flight.
Given: [latex]u = 14[/latex] m/s, [latex]\theta = 85°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(14)^2 \sin^2 85°}{2 \times 9.8}[/latex]
[latex]H = \frac{196 \times (0.996)^2}{19.6} = \frac{196 \times 0.992}{19.6} = \frac{194.43}{19.6} = 9.92[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(14)^2 \sin 170°}{9.8}[/latex]
[latex]R = \frac{196 \times 0.174}{9.8} = \frac{34.10}{9.8} = 3.48[/latex] m
Step 3: Time of Flight
[latex]T = \frac{2u \sin \theta}{g} = \frac{2 \times 14 \times \sin 85°}{9.8}[/latex]
[latex]T = \frac{28 \times 0.996}{9.8} = \frac{27.89}{9.8} = 2.85[/latex] s
A projectile is fired at 42° with initial velocity 38 m/s. Find the parameters.
Given: [latex]u = 38[/latex] m/s, [latex]\theta = 42°[/latex], [latex]g = 9.8[/latex] m/s²
Step 1: Maximum Height
[latex]H = \frac{u^2 \sin^2 \theta}{2g} = \frac{(38)^2 \sin^2 42°}{2 \times 9.8}[/latex]
[latex]H = \frac{1444 \times (0.669)^2}{19.6} = \frac{1444 \times 0.448}{19.6} = \frac{646.91}{19.6} = 33.01[/latex] m
Step 2: Range
[latex]R = \frac{u^2 \sin 2\theta}{g} = \frac{(38)^2 \sin 84°}{9.8}[/latex]
[latex]R = \frac{1444 \times 0.995}{9.8} = \frac{1436.78}{9.8} = 146.61[/latex] m
Step 3: Time of Flight [latex]T = \f